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Structure (mathematical logic)
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<p>In <a href="/facts/Universal_algebra/uMJAL6ER">universal algebra</a> and in <a href="/facts/Model_theory/vLWVV8sQ">model theory</a>, a structure consists of a <a href="/facts/Set_(mathematics)/BfucfHMq">set</a> along with a collection of <a href="/facts/Finitary/B0eHCDjw">finitary operations</a> and <a href="/facts/Finitary_relation/JkNnVYNj">relations</a> that are defined on it. </p><p>Universal algebra studies structures that generalize the <a href="/facts/Algebraic_structure/TkuXca24">algebraic structures</a> such as <a href="/facts/Group_(mathematics)/IQTYqINt">groups</a>, <a href="/facts/Ring_(mathematics)/JnCUXz63">rings</a>, <a href="/facts/Field_(mathematics)/xAjAS4ko">fields</a> and <a href="/facts/Vector_space/M1pxMLx2">vector spaces</a>. The term universal algebra is used for structures of <a href="/facts/First-order_logic/lcISqWIg">first-order theories</a> with no <a href="/facts/Relation_symbol/r9BwGgaK">relation symbols</a>. <a href="/facts/Model_theory/vLWVV8sQ">Model theory</a> has a different scope that encompasses more arbitrary <a href="/facts/First-order_logic/lcISqWIg">first-order theories</a>, including <a href="/facts/Foundations_of_mathematics/yjpbbUkt">foundational</a> structures such as models of <a href="/facts/Set_theory/1ptfsvUP">set theory</a>. </p><p>From the model-theoretic point of view, structures are the objects used to define the semantics of <a href="/facts/First-order_logic/lcISqWIg">first-order logic</a>, cf. also <a href="/facts/Semantic_theory_of_truth/BtXbu5nt">Tarski's theory of truth</a> or <a href="/facts/First-order_logic/lcISqWIg">Tarskian semantics</a>. </p><p>For a given theory in model theory, a structure is called a model if it satisfies the defining axioms of that theory, although it is sometimes disambiguated as a <i><a href="/facts/Semantic_model/hgysE2kP">semantic model</a></i> when one discusses the notion in the more general setting of <a href="/facts/Mathematical_model/CGhyuxgz">mathematical models</a>. Logicians sometimes refer to structures as "<a href="/facts/Interpretation_(logic)/goPmjjQG">interpretations</a>", whereas the term "interpretation" generally has a different (although related) meaning in model theory; see <a href="/facts/Interpretation_(model_theory)/YlCGQvsq">interpretation (model theory)</a>. </p><p>In <a href="/facts/Database_theory/unyQa3cK">database theory</a>, structures with no functions are studied as models for relational <a href="/facts/Database/VYcpkevb">databases</a>, in the form of <a href="/facts/Relational_model/VFf4dDIU">relational models</a>. </p>