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class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-No_footnotes plainlinks metadata ambox ambox-style ambox-No_footnotes" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/40px-Text_document_with_red_question_mark.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/60px-Text_document_with_red_question_mark.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/80px-Text_document_with_red_question_mark.svg.png 2x" data-file-width="48" data-file-height="48" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article includes a <a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources">list of references</a>, <a href="/wiki/Wikipedia:Further_reading" title="Wikipedia:Further reading">related reading</a>, or <a href="/wiki/Wikipedia:External_links" title="Wikipedia:External links">external links</a>, <b>but its sources remain unclear because it lacks <a href="/wiki/Wikipedia:Citing_sources#Inline_citations" title="Wikipedia:Citing sources">inline citations</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Wikipedia:WikiProject_Fact_and_Reference_Check" class="mw-redirect" title="Wikipedia:WikiProject Fact and Reference Check">improve</a> this article by <a href="/wiki/Wikipedia:When_to_cite" title="Wikipedia:When to cite">introducing</a> more precise citations.</span> <span class="date-container"><i>(<span class="date">June 2022</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Branch of mathematics relating to posets</div> <p><b>Domain theory</b> is a branch of <a href="/wiki/Mathematics" title="Mathematics">mathematics</a> that studies special kinds of <a href="/wiki/Partially_ordered_set" title="Partially ordered set">partially ordered sets</a> (posets) commonly called <b>domains</b>. Consequently, domain theory can be considered as a branch of <a href="/wiki/Order_theory" title="Order theory">order theory</a>. The field has major applications in <a href="/wiki/Computer_science" title="Computer science">computer science</a>, where it is used to specify <a href="/wiki/Denotational_semantics" title="Denotational semantics">denotational semantics</a>, especially for <a href="/wiki/Functional_programming" title="Functional programming">functional programming languages</a>. Domain theory formalizes the intuitive ideas of approximation and convergence in a very general way and is closely related to <a href="/wiki/Topology" title="Topology">topology</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Motivation_and_intuition">Motivation and intuition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Domain_theory&action=edit&section=1" title="Edit section: Motivation and intuition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The primary motivation for the study of domains, which was initiated by <a href="/wiki/Dana_Scott" title="Dana Scott">Dana Scott</a> in the late 1960s, was the search for a <a href="/wiki/Denotational_semantics" title="Denotational semantics">denotational semantics</a> of the <a href="/wiki/Lambda_calculus" title="Lambda calculus">lambda calculus</a>. In this formalism, one considers "functions" specified by certain terms in the language. In a purely <a href="/wiki/Syntax" title="Syntax">syntactic</a> way, one can go from simple functions to functions that take other functions as their input arguments. Using again just the syntactic transformations available in this formalism, one can obtain so-called <a href="/wiki/Fixed-point_combinator" title="Fixed-point combinator">fixed-point combinators</a> (the best-known of which is the <a href="/wiki/Fixed-point_combinator#Y_combinator" title="Fixed-point combinator">Y combinator</a>); these, by definition, have the property that <i>f</i>(<b>Y</b>(<i>f</i>)) = <b>Y</b>(<i>f</i>) for all functions <i>f</i>. </p><p>To formulate such a denotational semantics, one might first try to construct a <i>model</i> for the lambda calculus, in which a genuine (total) function is associated with each lambda term. Such a model would formalize a link between the lambda calculus as a purely syntactic system and the lambda calculus as a notational system for manipulating concrete mathematical functions. The <a href="/wiki/Combinator_calculus" class="mw-redirect" title="Combinator calculus">combinator calculus</a> is such a model. However, the elements of the combinator calculus are functions from functions to functions; in order for the elements of a model of the lambda calculus to be of arbitrary domain and range, they could not be true functions, only <a href="/wiki/Partial_functions" class="mw-redirect" title="Partial functions">partial functions</a>. </p><p>Scott got around this difficulty by formalizing a notion of "partial" or "incomplete" information to represent computations that have not yet returned a result. This was modeled by considering, for each domain of computation (e.g. the natural numbers), an additional element that represents an <i>undefined</i> output, i.e. the "result" of a computation that never ends. In addition, the domain of computation is equipped with an <i>ordering relation</i>, in which the "undefined result" is the <a href="/wiki/Least_element" class="mw-redirect" title="Least element">least element</a>. </p><p>The important step to finding a model for the lambda calculus is to consider only those functions (on such a partially ordered set) that are guaranteed to have <a href="/wiki/Least_fixed_point" title="Least fixed point">least fixed points</a>. The set of these functions, together with an appropriate ordering, is again a "domain" in the sense of the theory. But the restriction to a subset of all available functions has another great benefit: it is possible to obtain domains that contain their own <a href="/wiki/Function_space" title="Function space">function spaces</a>, i.e. one gets functions that can be applied to themselves. </p><p>Beside these desirable properties, domain theory also allows for an appealing intuitive interpretation. As mentioned above, the domains of computation are always partially ordered. This ordering represents a hierarchy of information or knowledge. The higher an element is within the order, the more specific it is and the more information it contains. Lower elements represent incomplete knowledge or intermediate results. </p><p>Computation then is modeled by applying <a href="/wiki/Monotonic" class="mw-redirect" title="Monotonic">monotone</a> <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">functions</a> repeatedly on elements of the domain in order to refine a result. Reaching a <a href="/wiki/Fixed_point_(mathematics)" title="Fixed point (mathematics)">fixed point</a> is equivalent to finishing a calculation. Domains provide a superior setting for these ideas since fixed points of monotone functions can be guaranteed to exist and, under additional restrictions, can be approximated from below. </p> <div class="mw-heading mw-heading2"><h2 id="A_guide_to_the_formal_definitions">A guide to the formal definitions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Domain_theory&action=edit&section=2" title="Edit section: A guide to the formal definitions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In this section, the central concepts and definitions of domain theory will be introduced. The above intuition of domains being <i>information orderings</i> will be emphasized to motivate the mathematical formalization of the theory. The precise formal definitions are to be found in the dedicated articles for each concept. A list of general order-theoretic definitions, which include domain theoretic notions as well can be found in the <a href="/wiki/Order_theory_glossary" class="mw-redirect" title="Order theory glossary">order theory glossary</a>. The most important concepts of domain theory will nonetheless be introduced below. </p> <div class="mw-heading mw-heading3"><h3 id="Directed_sets_as_converging_specifications">Directed sets as converging specifications</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Domain_theory&action=edit&section=3" title="Edit section: Directed sets as converging specifications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>As mentioned before, domain theory deals with <a href="/wiki/Partially_ordered_set" title="Partially ordered set">partially ordered sets</a> to model a domain of computation. The goal is to interpret the elements of such an order as <i>pieces of information</i> or <i>(partial) results of a computation</i>, where elements that are higher in the order extend the information of the elements below them in a consistent way. From this simple intuition it is already clear that domains often do not have a <a href="/wiki/Greatest_element" class="mw-redirect" title="Greatest element">greatest element</a>, since this would mean that there is an element that contains the information of <i>all</i> other elements—a rather uninteresting situation. </p><p>A concept that plays an important role in the theory is that of a <b><a href="/wiki/Directed_set" title="Directed set">directed subset</a></b> of a domain; a directed subset is a non-empty subset of the order in which any two elements have an <a href="/wiki/Upper_bound" class="mw-redirect" title="Upper bound">upper bound</a> that is an element of this subset. In view of our intuition about domains, this means that any two pieces of information within the directed subset are <i>consistently</i> extended by some other element in the subset. Hence we can view directed subsets as <i>consistent specifications</i>, i.e. as sets of partial results in which no two elements are contradictory. This interpretation can be compared with the notion of a <a href="/wiki/Convergent_sequence" class="mw-redirect" title="Convergent sequence">convergent sequence</a> in <a href="/wiki/Mathematical_analysis" title="Mathematical analysis">analysis</a>, where each element is more specific than the preceding one. Indeed, in the theory of <a href="/wiki/Metric_space" title="Metric space">metric spaces</a>, sequences play a role that is in many aspects analogous to the role of directed sets in domain theory. </p><p>Now, as in the case of sequences, we are interested in the <i>limit</i> of a directed set. According to what was said above, this would be an element that is the most general piece of information that extends the information of all elements of the directed set, i.e. the unique element that contains <i>exactly</i> the information that was present in the directed set, and nothing more. In the formalization of order theory, this is just the <b><a href="/wiki/Least_upper_bound" class="mw-redirect" title="Least upper bound">least upper bound</a></b> of the directed set. As in the case of the limit of a sequence, the least upper bound of a directed set does not always exist. </p><p>Naturally, one has a special interest in those domains of computations in which all consistent specifications <i>converge</i>, i.e. in orders in which all directed sets have a least upper bound. This property defines the class of <b><a href="/wiki/Directed_complete_partial_order" class="mw-redirect" title="Directed complete partial order">directed-complete partial orders</a></b>, or <b>dcpo</b> for short. Indeed, most considerations of domain theory do only consider orders that are at least directed complete. </p><p>From the underlying idea of partially specified results as representing incomplete knowledge, one derives another desirable property: the existence of a <b><a href="/wiki/Least_element" class="mw-redirect" title="Least element">least element</a></b>. Such an element models that state of no information—the place where most computations start. It also can be regarded as the output of a computation that does not return any result at all. </p> <div class="mw-heading mw-heading3"><h3 id="Computations_and_domains">Computations and domains</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Domain_theory&action=edit&section=4" title="Edit section: Computations and domains"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Now that we have some basic formal descriptions of what a domain of computation should be, we can turn to the computations themselves. Clearly, these have to be functions, taking inputs from some computational domain and returning outputs in some (possibly different) domain. However, one would also expect that the output of a function will contain more information when the information content of the input is increased. Formally, this means that we want a function to be <b><a href="/wiki/Monotonic" class="mw-redirect" title="Monotonic">monotonic</a></b>. </p><p>When dealing with <b><a href="/wiki/Complete_partial_order" title="Complete partial order">dcpos</a></b>, one might also want computations to be compatible with the formation of limits of a directed set. Formally, this means that, for some function <i>f</i>, the image <i>f</i>(<i>D</i>) of a directed set <i>D</i> (i.e. the set of the images of each element of <i>D</i>) is again directed and has as a least upper bound the image of the least upper bound of <i>D</i>. One could also say that <i>f</i> <i>preserves directed suprema</i>. Also note that, by considering directed sets of two elements, such a function also has to be monotonic. These properties give rise to the notion of a <b><a href="/wiki/Scott-continuous" class="mw-redirect" title="Scott-continuous">Scott-continuous</a></b> function. Since this often is not ambiguous one also may speak of <i>continuous functions</i>. </p> <div class="mw-heading mw-heading3"><h3 id="Approximation_and_finiteness">Approximation and finiteness</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Domain_theory&action=edit&section=5" title="Edit section: Approximation and finiteness"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Domain theory is a purely <i>qualitative</i> approach to modeling the structure of information states. One can say that something contains more information, but the amount of additional information is not specified. Yet, there are some situations in which one wants to speak about elements that are in a sense much simpler (or much more incomplete) than a given state of information. For example, in the natural subset-inclusion ordering on some <a href="/wiki/Powerset" class="mw-redirect" title="Powerset">powerset</a>, any infinite element (i.e. set) is much more "informative" than any of its <i>finite</i> subsets. </p><p>If one wants to model such a relationship, one may first want to consider the induced strict order < of a domain with order ≤. However, while this is a useful notion in the case of total orders, it does not tell us much in the case of partially ordered sets. Considering again inclusion-orders of sets, a set is already strictly smaller than another, possibly infinite, set if it contains just one less element. One would, however, hardly agree that this captures the notion of being "much simpler". </p> <div class="mw-heading mw-heading3"><h3 id="Way-below_relation">Way-below relation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Domain_theory&action=edit&section=6" title="Edit section: Way-below relation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A more elaborate approach leads to the definition of the so-called <b>order of approximation</b>, which is more suggestively also called the <b>way-below relation</b>. An element <i>x</i> is <i>way below</i> an element <i>y</i>, if, for every directed set <i>D</i> with supremum such that </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\sqsubseteq \sup D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>⊑</mo> <mo movablelimits="true" form="prefix">sup</mo> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y\sqsubseteq \sup D}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77a179fe8eecc2594cfbb26ad95a8bb08eebd73b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.067ex; height:2.509ex;" alt="{\displaystyle y\sqsubseteq \sup D}"></span>,</dd></dl> <p>there is some element <i>d</i> in <i>D</i> such that </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\sqsubseteq d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>⊑</mo> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\sqsubseteq d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d5978c94945aa63b913b9324e5fa0e1e4e9bb64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.644ex; height:2.343ex;" alt="{\displaystyle x\sqsubseteq d}"></span>.</dd></dl> <p>Then one also says that <i>x</i> <i>approximates</i> <i>y</i> and writes </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\ll y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>≪</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\ll y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa24e30006077de6c6e962b966f9122dd996df6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.099ex; height:2.176ex;" alt="{\displaystyle x\ll y}"></span>.</dd></dl> <p>This does imply that </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\sqsubseteq y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>⊑</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\sqsubseteq y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f4a92535bc2b9fd7b96564ffac7cd5f0e5510a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.584ex; height:2.343ex;" alt="{\displaystyle x\sqsubseteq y}"></span>,</dd></dl> <p>since the singleton set {<i>y</i>} is directed. For an example, in an ordering of sets, an infinite set is way above any of its finite subsets. On the other hand, consider the directed set (in fact, the <a href="/wiki/Total_order#Chains" title="Total order">chain</a>) of finite sets </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{0\},\{0,1\},\{0,1,2\},\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{0\},\{0,1\},\{0,1,2\},\ldots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bdd36ffef38d2eb09721ebeed764f58ea07d458" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.876ex; height:2.843ex;" alt="{\displaystyle \{0\},\{0,1\},\{0,1,2\},\ldots }"></span></dd></dl> <p>Since the supremum of this chain is the set of all natural numbers <b>N</b>, this shows that no infinite set is way below <b>N</b>. </p><p>However, being way below some element is a <i>relative</i> notion and does not reveal much about an element alone. For example, one would like to characterize finite sets in an order-theoretic way, but even infinite sets can be way below some other set. The special property of these <b>finite</b> elements <i>x</i> is that they are way below themselves, i.e. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\ll x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>≪</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\ll x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3039a4d48110b3e4db7319efa3b2b6e8e8932bfd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.273ex; height:1.843ex;" alt="{\displaystyle x\ll x}"></span>.</dd></dl> <p>An element with this property is also called <b><a href="/wiki/Compact_element" title="Compact element">compact</a></b>. Yet, such elements do not have to be "finite" nor "compact" in any other mathematical usage of the terms. The notation is nonetheless motivated by certain parallels to the respective notions in <a href="/wiki/Set_theory" title="Set theory">set theory</a> and <a href="/wiki/Topology" title="Topology">topology</a>. The compact elements of a domain have the important special property that they cannot be obtained as a limit of a directed set in which they did not already occur. </p><p>Many other important results about the way-below relation support the claim that this definition is appropriate to capture many important aspects of a domain. </p> <div class="mw-heading mw-heading3"><h3 id="Bases_of_domains">Bases of domains</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Domain_theory&action=edit&section=7" title="Edit section: Bases of domains"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The previous thoughts raise another question: is it possible to guarantee that all elements of a domain can be obtained as a limit of much simpler elements? This is quite relevant in practice, since we cannot compute infinite objects but we may still hope to approximate them arbitrarily closely. </p><p>More generally, we would like to restrict to a certain subset of elements as being sufficient for getting all other elements as least upper bounds. Hence, one defines a <b>base</b> of a poset <i>P</i> as being a subset <i>B</i> of <i>P</i>, such that, for each <i>x</i> in <i>P</i>, the set of elements in <i>B</i> that are way below <i>x</i> contains a directed set with supremum <i>x</i>. The poset <i>P</i> is a <b>continuous poset</b> if it has some base. Especially, <i>P</i> itself is a base in this situation. In many applications, one restricts to continuous (d)cpos as a main object of study. </p><p>Finally, an even stronger restriction on a partially ordered set is given by requiring the existence of a base of <i>finite</i> elements. Such a poset is called <b><a href="/wiki/Algebraic_poset" class="mw-redirect" title="Algebraic poset">algebraic</a></b>. From the viewpoint of denotational semantics, algebraic posets are particularly well-behaved, since they allow for the approximation of all elements even when restricting to finite ones. As remarked before, not every finite element is "finite" in a classical sense and it may well be that the finite elements constitute an <a href="/wiki/Uncountable" class="mw-redirect" title="Uncountable">uncountable</a> set. </p><p>In some cases, however, the base for a poset is <a href="/wiki/Countable" class="mw-redirect" title="Countable">countable</a>. In this case, one speaks of an <b>ω-continuous</b> poset. Accordingly, if the countable base consists entirely of finite elements, we obtain an order that is <b>ω-algebraic</b>. </p> <div class="mw-heading mw-heading3"><h3 id="Special_types_of_domains">Special types of domains</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Domain_theory&action=edit&section=8" title="Edit section: Special types of domains"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A simple special case of a domain is known as an <b>elementary</b> or <b>flat domain</b>. This consists of a set of incomparable elements, such as the integers, along with a single "bottom" element considered smaller than all other elements. </p><p>One can obtain a number of other interesting special classes of ordered structures that could be suitable as "domains". We already mentioned continuous posets and algebraic posets. More special versions of both are continuous and algebraic <a href="/wiki/Complete_partial_order" title="Complete partial order">cpos</a>. Adding even further <a href="/wiki/Completeness_(order_theory)" title="Completeness (order theory)">completeness properties</a> one obtains <a href="/wiki/Lattice_(order)#Continuity_and_algebraicity" title="Lattice (order)">continuous lattices</a> and <a href="/wiki/Algebraic_lattices" class="mw-redirect" title="Algebraic lattices">algebraic lattices</a>, which are just <a href="/wiki/Complete_lattice" title="Complete lattice">complete lattices</a> with the respective properties. For the algebraic case, one finds broader classes of posets that are still worth studying: historically, the <a href="/wiki/Scott_domain" title="Scott domain">Scott domains</a> were the first structures to be studied in domain theory. Still wider classes of domains are constituted by <a href="/w/index.php?title=SFP-domain&action=edit&redlink=1" class="new" title="SFP-domain (page does not exist)">SFP-domains</a>, <a href="/w/index.php?title=L-domain&action=edit&redlink=1" class="new" title="L-domain (page does not exist)">L-domains</a>, and <a href="/w/index.php?title=Bifinite_domain&action=edit&redlink=1" class="new" title="Bifinite domain (page does not exist)">bifinite domains</a>. </p><p>All of these classes of orders can be cast into various <a href="/wiki/Category_(mathematics)" title="Category (mathematics)">categories</a> of dcpos, using functions that are monotone, Scott-continuous, or even more specialized as <a href="/wiki/Morphism" title="Morphism">morphisms</a>. Finally, note that the term <i>domain</i> itself is not exact and thus is only used as an abbreviation when a formal definition has been given before or when the details are irrelevant. </p> <div class="mw-heading mw-heading2"><h2 id="Important_results">Important results</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Domain_theory&action=edit&section=9" title="Edit section: Important results"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A poset <i>D</i> is a dcpo if and only if each chain in <i>D</i> has a supremum. (The 'if' direction relies on the <a href="/wiki/Axiom_of_choice" title="Axiom of choice">axiom of choice</a>.) </p><p>If <i>f</i> is a continuous function on a domain <i>D</i> then it has a least fixed point, given as the least upper bound of all finite iterations of <i>f</i> on the least element ⊥: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {fix} (f)=\bigsqcup _{n\in \mathbb {N} }f^{n}(\bot )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>fix</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo>⨆</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </munder> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi mathvariant="normal">⊥</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {fix} (f)=\bigsqcup _{n\in \mathbb {N} }f^{n}(\bot )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e51b9c1014cf355a545432e4b5f1ab64c996ee5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:18.585ex; height:5.676ex;" alt="{\displaystyle \operatorname {fix} (f)=\bigsqcup _{n\in \mathbb {N} }f^{n}(\bot )}"></span>.</dd></dl> <p>This is the <a href="/wiki/Kleene_fixed-point_theorem" title="Kleene fixed-point theorem">Kleene fixed-point theorem</a>. The <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sqcup }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊔</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sqcup }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1596aedf354da694149e44ce2bf53ede54eca8cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \sqcup }"></span> symbol is the <a href="/wiki/Join_and_meet" title="Join and meet">directed join</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Generalizations">Generalizations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Domain_theory&action=edit&section=10" title="Edit section: Generalizations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <a href="/wiki/Continuity_space" class="mw-redirect" title="Continuity space">continuity space</a> is a generalization of metric spaces and <a href="/wiki/Poset" class="mw-redirect" title="Poset">posets</a> that can be used to unify the notions of metric spaces and domains. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Domain_theory&action=edit&section=11" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Category_theory" title="Category theory">Category theory</a></li> <li><a href="/wiki/Denotational_semantics" title="Denotational semantics">Denotational semantics</a></li> <li><a href="/wiki/Scott_domain" title="Scott domain">Scott domain</a></li> <li><a href="/wiki/Scott_information_system" title="Scott information system">Scott information system</a></li> <li><a href="/wiki/Type_theory" title="Type theory">Type theory</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Domain_theory&action=edit&section=12" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFG._GierzK._H._HofmannK._KeimelJ._D._Lawson2003" class="citation encyclopaedia cs1">G. Gierz; K. H. Hofmann; K. Keimel; J. D. Lawson; M. Mislove; D. S. Scott (2003). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/continuouslattic0000unse">"Continuous Lattices and Domains"</a></span>. <i>Encyclopedia of Mathematics and its Applications</i>. Vol. 93. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-80338-1" title="Special:BookSources/0-521-80338-1"><bdi>0-521-80338-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Continuous+Lattices+and+Domains&rft.btitle=Encyclopedia+of+Mathematics+and+its+Applications&rft.pub=Cambridge+University+Press&rft.date=2003&rft.isbn=0-521-80338-1&rft.au=G.+Gierz&rft.au=K.+H.+Hofmann&rft.au=K.+Keimel&rft.au=J.+D.+Lawson&rft.au=M.+Mislove&rft.au=D.+S.+Scott&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fcontinuouslattic0000unse&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADomain+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSamson_Abramsky,_Achim_Jung1994" class="citation conference cs1"><a href="/wiki/Samson_Abramsky" title="Samson Abramsky">Samson Abramsky</a>, Achim Jung (1994). <a rel="nofollow" class="external text" href="http://www.cs.bham.ac.uk/~axj/pub/papers/handy1.pdf">"Domain theory"</a> <span class="cs1-format">(PDF)</span>. In S. Abramsky; <a href="/wiki/Dov_Gabbay" title="Dov Gabbay">D. M. Gabbay</a>; <a href="/wiki/Tom_Maibaum" title="Tom Maibaum">T. S. E. Maibaum</a> (eds.). <i>Handbook of Logic in Computer Science</i>. Vol. III. Oxford University Press. pp. <span class="nowrap">1–</span>168. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-19-853762-X" title="Special:BookSources/0-19-853762-X"><bdi>0-19-853762-X</bdi></a><span class="reference-accessdate">. Retrieved <span class="nowrap">2007-10-13</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=conference&rft.atitle=Domain+theory&rft.btitle=Handbook+of+Logic+in+Computer+Science&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1-%3C%2Fspan%3E168&rft.pub=Oxford+University+Press&rft.date=1994&rft.isbn=0-19-853762-X&rft.au=Samson+Abramsky%2C+Achim+Jung&rft_id=http%3A%2F%2Fwww.cs.bham.ac.uk%2F~axj%2Fpub%2Fpapers%2Fhandy1.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADomain+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAlex_Simpson2001–2002" class="citation book cs1">Alex Simpson (2001–2002). "Part III: Topological Spaces from a Computational Perspective". <a rel="nofollow" class="external text" href="https://web.archive.org/web/20050427104159/http://www.dcs.ed.ac.uk/home/als/Teaching/MSfS/l3.ps"><i>Mathematical Structures for Semantics</i></a>. Archived from <a rel="nofollow" class="external text" href="http://www.dcs.ed.ac.uk/home/als/Teaching/MSfS/l3.ps">the original</a> on 2005-04-27<span class="reference-accessdate">. Retrieved <span class="nowrap">2007-10-13</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Part+III%3A+Topological+Spaces+from+a+Computational+Perspective&rft.btitle=Mathematical+Structures+for+Semantics&rft.date=2001%2F2002&rft.au=Alex+Simpson&rft_id=http%3A%2F%2Fwww.dcs.ed.ac.uk%2Fhome%2Fals%2FTeaching%2FMSfS%2Fl3.ps&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADomain+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFD._S._Scott1975" class="citation book cs1"><a href="/wiki/Dana_Scott" title="Dana Scott">D. S. Scott</a> (1975). "Data types as lattices". In Müller, G.H.; Oberschelp, A.; Potthoff, K. (eds.). <i>ISILC Logic Conference</i>. Lecture Notes in Mathematics. Vol. 499. Springer-Verlag. pp. <span class="nowrap">579–</span>651. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBFb0079432">10.1007/BFb0079432</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-07534-9" title="Special:BookSources/978-3-540-07534-9"><bdi>978-3-540-07534-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Data+types+as+lattices&rft.btitle=ISILC+Logic+Conference&rft.series=Lecture+Notes+in+Mathematics&rft.pages=%3Cspan+class%3D%22nowrap%22%3E579-%3C%2Fspan%3E651&rft.pub=Springer-Verlag&rft.date=1975&rft_id=info%3Adoi%2F10.1007%2FBFb0079432&rft.isbn=978-3-540-07534-9&rft.au=D.+S.+Scott&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADomain+theory" class="Z3988"></span> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFScott1976" class="citation journal cs1">Scott, Dana (1976). "Data Types as Lattices". <i>SIAM Journal on Computing</i>. <b>5</b> (3): <span class="nowrap">522–</span>587. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1137%2F0205037">10.1137/0205037</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=SIAM+Journal+on+Computing&rft.atitle=Data+Types+as+Lattices&rft.volume=5&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E522-%3C%2Fspan%3E587&rft.date=1976&rft_id=info%3Adoi%2F10.1137%2F0205037&rft.aulast=Scott&rft.aufirst=Dana&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADomain+theory" class="Z3988"></span></li></ul></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCarl_A._Gunter1992" class="citation book cs1">Carl A. Gunter (1992). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Gu2wh2TqVcUC&q=%22domain+theory%22"><i>Semantics of Programming Languages</i></a>. MIT Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780262570954" title="Special:BookSources/9780262570954"><bdi>9780262570954</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Semantics+of+Programming+Languages&rft.pub=MIT+Press&rft.date=1992&rft.isbn=9780262570954&rft.au=Carl+A.+Gunter&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DGu2wh2TqVcUC%26q%3D%2522domain%2Btheory%2522&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADomain+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFB._A._DaveyH._A._Priestley2002" class="citation book cs1">B. A. Davey; <a href="/wiki/Hilary_Priestley" title="Hilary Priestley">H. A. Priestley</a> (2002). <a href="/wiki/Introduction_to_Lattices_and_Order" title="Introduction to Lattices and Order"><i>Introduction to Lattices and Order</i></a> (2nd ed.). Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-78451-4" title="Special:BookSources/0-521-78451-4"><bdi>0-521-78451-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Lattices+and+Order&rft.edition=2nd&rft.pub=Cambridge+University+Press&rft.date=2002&rft.isbn=0-521-78451-4&rft.au=B.+A.+Davey&rft.au=H.+A.+Priestley&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADomain+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCarl_HewittHenry_Baker1977" class="citation conference cs1">Carl Hewitt; Henry Baker (August 1977). <a rel="nofollow" class="external text" href="https://apps.dtic.mil/dtic/tr/fulltext/u2/a052266.pdf">"Actors and Continuous Functionals"</a> <span class="cs1-format">(PDF)</span>. <i>Proceedings of IFIP Working Conference on Formal Description of Programming Concepts</i>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20190412190831/https://apps.dtic.mil/dtic/tr/fulltext/u2/a052266.pdf">Archived</a> <span class="cs1-format">(PDF)</span> from the original on April 12, 2019.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=conference&rft.atitle=Actors+and+Continuous+Functionals&rft.btitle=Proceedings+of+IFIP+Working+Conference+on+Formal+Description+of+Programming+Concepts&rft.date=1977-08&rft.au=Carl+Hewitt&rft.au=Henry+Baker&rft_id=https%3A%2F%2Fapps.dtic.mil%2Fdtic%2Ftr%2Ffulltext%2Fu2%2Fa052266.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADomain+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFV._Stoltenberg-HansenI._LindstromE._R._Griffor1994" class="citation book cs1">V. Stoltenberg-Hansen; I. Lindstrom; E. R. Griffor (1994). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/mathematicaltheo0000stol"><i>Mathematical Theory of Domains</i></a></span>. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-38344-7" title="Special:BookSources/0-521-38344-7"><bdi>0-521-38344-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematical+Theory+of+Domains&rft.pub=Cambridge+University+Press&rft.date=1994&rft.isbn=0-521-38344-7&rft.au=V.+Stoltenberg-Hansen&rft.au=I.+Lindstrom&rft.au=E.+R.+Griffor&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fmathematicaltheo0000stol&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADomain+theory" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Domain_theory&action=edit&section=13" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://www.cs.nott.ac.uk/~gmh/domains.html">Introduction to Domain Theory</a> by Graham Hutton, <a href="/wiki/University_of_Nottingham" title="University of Nottingham">University of Nottingham</a></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Domain_theory&oldid=1272158172">https://en.wikipedia.org/w/index.php?title=Domain_theory&oldid=1272158172</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Domain_theory" title="Category:Domain theory">Domain theory</a></li><li><a href="/wiki/Category:Fixed_points_(mathematics)" title="Category:Fixed points (mathematics)">Fixed points (mathematics)</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_lacking_in-text_citations_from_June_2022" title="Category:Articles lacking in-text citations from June 2022">Articles lacking in-text citations from June 2022</a></li><li><a href="/wiki/Category:All_articles_lacking_in-text_citations" title="Category:All articles lacking in-text citations">All articles lacking in-text citations</a></li><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 27 January 2025, at 11:21<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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