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Partial Algebra
In abstract algebra, a partial algebra is a generalization of universal algebra to partial function, partial Operation (mathematics), operations. Example(s) * partial groupoid * Field (mathematics), field — the multiplicative inversion is the only proper partial operation * effect algebras 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] |
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, algebra meant ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Universal Algebra
Universal algebra (sometimes called general algebra) is the field of mathematics that studies algebraic structures themselves, not examples ("models") of algebraic structures. For instance, rather than take particular groups as the object of study, in universal algebra one takes the class of groups as an object of study. Basic idea In universal algebra, an algebra (or algebraic structure) is a set ''A'' together with a collection of operations on ''A''. An ''n''- ary operation on ''A'' is a function that takes ''n'' elements of ''A'' and returns a single element of ''A''. Thus, a 0-ary operation (or ''nullary operation'') can be represented simply as an element of ''A'', or a '' constant'', often denoted by a letter like ''a''. A 1-ary operation (or ''unary operation'') is simply a function from ''A'' to ''A'', often denoted by a symbol placed in front of its argument, like ~''x''. A 2-ary operation (or ''binary operation'') is often denoted by a symbol placed between its argum ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial Function
In mathematics, a partial function from a set to a set is a function from a subset of (possibly itself) to . The subset , that is, the domain of viewed as a function, is called the domain of definition of . If equals , that is, if is defined on every element in , then is said to be total. More technically, a partial function is a binary relation over two sets that associates every element of the first set to ''at most'' one element of the second set; it is thus a functional binary relation. It generalizes the concept of a (total) function by not requiring every element of the first set to be associated to ''exactly'' one element of the second set. A partial function is often used when its exact domain of definition is not known or difficult to specify. This is the case in calculus, where, for example, the quotient of two functions is a partial function whose domain of definition cannot contain the zeros of the denominator. For this reason, in calculus, and more gene ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Operation (mathematics)
In mathematics, an operation is a function which takes zero or more input values (also called "''operands''" or "arguments") to a well-defined output value. The number of operands is the arity of the operation. The most commonly studied operations are binary operations (i.e., operations of arity 2), such as addition and multiplication, and unary operations (i.e., operations of arity 1), such as additive inverse and multiplicative inverse. An operation of arity zero, or nullary operation, is a constant. The mixed product is an example of an operation of arity 3, also called ternary operation. Generally, the arity is taken to be finite. However, infinitary operations are sometimes considered, in which case the "usual" operations of finite arity are called finitary operations. A partial operation is defined similarly to an operation, but with a partial function in place of a function. Types of operation There are two common types of operations: unary and binar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. 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 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 f ... 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 struct ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Field (mathematics)
In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do. A field is thus a fundamental algebraic structure which is widely used in algebra, number theory, and many other areas of mathematics. The best known fields are the field of rational numbers, the field of real numbers and the field of complex numbers. Many other fields, such as fields of rational functions, algebraic function fields, algebraic number fields, and ''p''-adic fields are commonly used and studied in mathematics, particularly in number theory and algebraic geometry. Most cryptographic protocols rely on finite fields, i.e., fields with finitely many elements. The relation of two fields is expressed by the notion of a field extension. Galois theory, initiated by Évariste Galois in the 1830s, is devoted to understanding the symmetries of field extensions. Among other results, thi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 physics or mathematics in the late 1980s and early 1990s. Since then, their mathematical properties and physical as well as computational significance have been studied by researchers in theoretical physics, mathematics and computer science. History In 1989, Giuntini and Greuling introduced structures for studying ''unsharp properties'', meaning those quantum events whose probability of occurring is strictly between zero and one (and is thus not an either-or event).Foulis, David J. "A Half-Century of Quantum Logic. What Have We Learned?" ''in'' Aerts, Diederik (ed.); Pykacz, Jarosław (ed.) ''Quantum Structures and the Nature of Reality.'' Springer, Dordrecht 1999. ISBN 978-94-017-2834-8. https://doi.org/10.1007/978-94-017-2834-8. In 19 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
