- Reduct
In

universal algebra and inmodel theory , a**reduct**of an algebraic structure is obtained by omitting some of the operations and relations of that structure. The converse of "reduct" is "expansion."**Definition**Let "A" be an

algebraic structure (in the sense ofuniversal algebra ) or equivalently a structure in the sense ofmodel theory , organized as a set "X" together with an indexedfamily ofoperation s and relations φ_{i}on that set, withindex set "I". Then the**reduct**of "A" defined by a subset "J" of "I" is the structure consisting of the set "X" and "J"-indexed family of operations and relations whose "j"-th operation or relation for "j"∈"J" is the "j"-th operation or relation of "A". That is, this reduct is the structure "A" with the omission of those operations and relations φ_{"i"}for which "i" is not in "J".Structure "A" is an

**expansion**of "B" just when "B" is a reduct of "A". That is, reduct and expansion are mutual converses.**Examples**The

monoid (**Z**, +, 0) ofinteger s underaddition is a reduct of the group (**Z**, +, −, 0) of integers under addition and negation, obtained by omitting negation.Conversely the group (

**Z**, +, −, 0) is the expansion of the monoid (**Z**, +, 0), expanding it with the operation of negation.**References***

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**Reduct**— Re*duct (r[ e]*d[u^]kt ), v. t. [L. reductus, p. p. of reducere. See {Reduce}.] To reduce. [Obs.] W. Warde. [1913 Webster] … The Collaborative International Dictionary of English**reduct**— rə̇ˈdəkt transitive verb ( ed/ ing/ s) Etymology: Latin reductus, past participle of reducere more at reduce 1. : reduce 2. dialect : deduct … Useful english dictionary**reduct**— /ri dukt /, v.t. to reduce. [ < L reductus, ptp. of reducere; see REDUCE] * * * … Universalium**reduct**— 1. noun A reducing agent. 2. verb a) To duct tape again. b) To channel through a duct again … Wiktionary**reduct**— re·duct … English syllables**gol-1, lengthened grade gōl-, reduct.st. gol-**— gol 1, lengthened grade gōl , reduct.st. gol English meaning: to lie; den (of animals) Deutsche Übersetzung: “liegen; Lager, Tierlager” Material: Arm. kalaɫ “cave, hiding place, nook, bolt hole or lair of a wild animal”; Gk.… … Proto-Indo-European etymological dictionary**Stable model semantics**— The concept of a stable model, or answer set, is used to define a declarative semantics for logic programs with negation as failure. This is one of several standard approaches to the meaning of negation in logic programming, along with program… … Wikipedia**Model theory**— This article is about the mathematical discipline. For the informal notion in other parts of mathematics and science, see Mathematical model. In mathematics, model theory is the study of (classes of) mathematical structures (e.g. groups, fields,… … Wikipedia**Pseudoelementary class**— In logic, a pseudoelementary class is a class of structures derived from an elementary class (one definable in first order logic) by omitting some of its sorts and relations. It is the mathematical logic counterpart of the notion in category… … Wikipedia**Lambda calculus**— In mathematical logic and computer science, lambda calculus, also written as λ calculus, is a formal system designed to investigate function definition, function application and recursion. It was introduced by Alonzo Church and Stephen Cole… … Wikipedia