In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type. The word homomorphism comes from the Ancient Greek language: ὁμός meaning "same" and μορφή meaning "form" or "shape". However, the word was apparently introduced to mathematics due to a (mis)translation of German ähnlich meaning "similar" to ὁμός meaning "same". The term "homomorphism" appeared as early as 1892, when it was attributed to the German mathematician Felix Klein (1849–1925).
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is derived from Ancient Greek ἴσος (isos) 'equal' and μορφή (morphe) 'form, shape'.
In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0.
In mathematical logic, model theory is the study of the relationship between formal theories, and their models. The aspects investigated include the number and size of models of a theory, the relationship of different models to each other, and their interaction with the formal language itself. In particular, model theorists also investigate the sets that can be defined in a model of a theory, and the relationship of such definable sets to each other. As a separate discipline, model theory goes back to Alfred Tarski, who first used the term "Theory of Models" in publication in 1954. Since the 1970s, the subject has been shaped decisively by Saharon Shelah's stability theory.
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it.
In mathematics, a subalgebra is a subset of an algebra, closed under all its operations, and carrying the induced operations.
In algebra, the kernel of a homomorphism is generally the inverse image of 0. An important special case is the kernel of a linear map. The kernel of a matrix, also called the null space, is the kernel of the linear map defined by the matrix.
In mathematics, specifically abstract algebra, the isomorphism theorems are theorems that describe the relationship among quotients, homomorphisms, and subobjects. Versions of the theorems exist for groups, rings, vector spaces, modules, Lie algebras, and other algebraic structures. In universal algebra, the isomorphism theorems can be generalized to the context of algebras and congruences.
In abstract algebra, a congruence relation is an equivalence relation on an algebraic structure that is compatible with the structure in the sense that algebraic operations done with equivalent elements will yield equivalent elements. Every congruence relation has a corresponding quotient structure, whose elements are the equivalence classes for the relation.
In mathematics, a subset of a given set is closed under an operation of the larger set if performing that operation on members of the subset always produces a member of that subset. For example, the natural numbers are closed under addition, but not under subtraction: 1 − 2 is not a natural number, although both 1 and 2 are.
In mathematical logic, the Löwenheim–Skolem theorem is a theorem on the existence and cardinality of models, named after Leopold Löwenheim and Thoralf Skolem.
In model theory, a branch of mathematical logic, two structures M and N of the same signature σ are called elementarily equivalent if they satisfy the same first-order σ-sentences.
In universal algebra and in model theory, a structure consists of a set along with a collection of finitary operations and relations that are defined on it.
In the mathematical field of graph theory, the Rado graph, Erdős–Rényi graph, or random graph is a countably infinite graph that can be constructed by choosing independently at random for each pair of its vertices whether to connect the vertices by an edge. The names of this graph honor Richard Rado, Paul Erdős, and Alfréd Rényi, mathematicians who studied it in the early 1960s; it appears even earlier in the work of Wilhelm Ackermann. The Rado graph can also be constructed non-randomly, by symmetrizing the membership relation of the hereditarily finite sets, by applying the BIT predicate to the binary representations of the natural numbers, or as an infinite Paley graph that has edges connecting pairs of prime numbers congruent to 1 mod 4 that are quadratic residues modulo each other.
In logic, especially mathematical logic, a signature lists and describes the non-logical symbols of a formal language. In universal algebra, a signature lists the operations that characterize an algebraic structure. In model theory, signatures are used for both purposes. They are rarely made explicit in more philosophical treatments of logic.
Fuzzy subalgebras theory is a chapter of fuzzy set theory. It is obtained from an interpretation in a multi-valued logic of axioms usually expressing the notion of subalgebra of a given algebraic structure.
In mathematics, a subring of a ring R is a subset of R that is itself a ring when binary operations of addition and multiplication on R are restricted to the subset, and that shares the same multiplicative identity as R.
In ring theory and Frobenius algebra extensions, areas of mathematics, there is a notion of depth two subring or depth of a Frobenius extension. The notion of depth two is important in a certain noncommutative Galois theory, which generates Hopf algebroids in place of the more classical Galois groups, whereas the notion of depth greater than two measures the defect, or distance, from being depth two in a tower of iterated endomorphism rings above the subring. A more recent definition of depth of any unital subring in any associative ring is proposed in a paper studying the depth of a subgroup of a finite group as group algebras over a commutative ring.
