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Partial Groupoid
In abstract algebra, a partial groupoid (also called halfgroupoid, pargoid, or partial magma) is a set endowed with a partial binary operation. A partial groupoid is a partial algebra In abstract algebra, a partial algebra is a generalization of universal algebra to partial operations. Example(s) * partial groupoid * field — the multiplicative inversion is the only proper partial operation * effect algebra Effect algebras .... Partial semigroup A partial groupoid (G,\circ) is called a partial semigroup (also called semigroupoid, semicategory, naked category, or precategory) if the following associative law holds: For all x,y,z \in G such that x\circ y\in G and y\circ z\in G, the following two statements hold: # x \circ (y \circ z) \in G if and only if ( x \circ y) \circ z \in G, and # x \circ (y \circ z ) = ( x \circ y) \circ z if x \circ (y \circ z) \in G (and, because of 1., also ( x \circ y) \circ z \in G). References Further reading * Algebraic stru ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Abstract Algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ''abstract algebra'' was coined in the early 20th century to distinguish this area of study from older parts of algebra, and more specifically from elementary algebra, the use of variables to represent numbers in computation and reasoning. Algebraic structures, with their associated homomorphisms, form mathematical categories. Category theory is a formalism that allows a unified way for expressing properties and constructions that are similar for various structures. Universal algebra is a related subject that studies types of algebraic structures as single objects. For example, the structure of groups is a single object in universal algebra, which is called the '' variety of groups''. History Before the nineteenth century, alge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Magma (algebra)
In abstract algebra, a magma, binar, or, rarely, groupoid is a basic kind of algebraic structure. Specifically, a magma consists of a set equipped with a single binary operation that must be closed by definition. No other properties are imposed. History and terminology The term ''groupoid'' was introduced in 1927 by Heinrich Brandt describing his Brandt groupoid (translated from the German ). The term was then appropriated by B. A. Hausmann and Øystein Ore (1937) in the sense (of a set with a binary operation) used in this article. In a couple of reviews of subsequent papers in Zentralblatt, Brandt strongly disagreed with this overloading of terminology. The Brandt groupoid is a groupoid in the sense used in category theory, but not in the sense used by Hausmann and Ore. Nevertheless, influential books in semigroup theory, including Clifford and Preston (1961) and Howie (1995) use groupoid in the sense of Hausmann and Ore. Hollings (2014) writes that the term ''grou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Binary Operation
In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two. More specifically, an internal binary operation ''on a set'' is a binary operation whose two domains and the codomain are the same set. Examples include the familiar arithmetic operations of addition, subtraction, and multiplication. Other examples are readily found in different areas of mathematics, such as vector addition, matrix multiplication, and conjugation in groups. An operation of arity two that involves several sets is sometimes also called a ''binary operation''. For example, scalar multiplication of vector spaces takes a scalar and a vector to produce a vector, and scalar product takes two vectors to produce a scalar. Such binary operations may be called simply binary functions. Binary operations are the keystone of most algebraic structures that are s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Partial Algebra
In abstract algebra, a partial algebra is a generalization of universal algebra to partial operations. Example(s) * partial groupoid * field — the multiplicative inversion is the only proper partial operation * effect algebra Effect algebras are partial algebras which abstract the (partial) algebraic properties of events that can be observed in quantum mechanics. Structures equivalent to effect algebras were introduced by three different research groups in theoretical ...s Structure There is a "Meta Birkhoff Theorem" by Andreka, Nemeti and Sain (1982). References Further reading * * * Algebraic structures {{algebra-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Semigroupoid
In mathematics, a semigroupoid (also called semicategory, naked category or precategory) is a partial algebra that satisfies the axioms for a smallSee e.g. , which requires the objects of a semigroupoid to form a set. Category (mathematics), category, except possibly for the requirement that there be an identity at each object. Semigroupoids generalise semigroups in the same way that small categories generalise monoids and groupoids generalise group (mathematics), groups. Semigroupoids have applications in the structural theory of semigroups. Formally, a ''semigroupoid'' consists of: * a Set (mathematics), set of things called ''objects''. * for every two objects ''A'' and ''B'' a set Mor(''A'',''B'') of things called ''morphisms from A to B''. If ''f'' is in Mor(''A'',''B''), we write ''f'' : ''A'' → ''B''. * for every three objects ''A'', ''B'' and ''C'' a binary operation Mor(''A'',''B'') × Mor(''B'',''C'') → Mor(''A'',''C'') called ''composition of morphisms''. The composit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Associative Law
In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement for expressions in logical proofs. Within an expression containing two or more occurrences in a row of the same associative operator, the order in which the operations are performed does not matter as long as the sequence of the operands is not changed. That is (after rewriting the expression with parentheses and in infix notation if necessary), rearranging the parentheses in such an expression will not change its value. Consider the following equations: \begin (2 + 3) + 4 &= 2 + (3 + 4) = 9 \,\\ 2 \times (3 \times 4) &= (2 \times 3) \times 4 = 24 . \end Even though the parentheses were rearranged on each line, the values of the expressions were not altered. Since this holds true when performing addition and multiplication on any real ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |