|
S4 Algebra
In abstract algebra, an interior algebra is a certain type of algebraic structure that encodes the idea of the topological interior of a set. Interior algebras are to topology and the modal logic S4 what Boolean algebras are to set theory and ordinary propositional logic. Interior algebras form a variety of modal algebras. Definition An interior algebra is an algebraic structure with the signature :⟨''S'', ·, +, ′, 0, 1, I⟩ where :⟨''S'', ·, +, ′, 0, 1⟩ is a Boolean algebra and postfix I designates a unary operator, the interior operator, satisfying the identities: # ''x''I ≤ ''x'' # ''x''II = ''x''I # (''xy'')I = ''x''I''y''I # 1I = 1 ''x''I is called the interior of ''x''. The dual of the interior operator is the closure operator C defined by ''x''C = ((''x''′)I)′. ''x''C is called the closure of ''x''. By the principle of duality, the closure operator satisfies the identities: # ''x''C ≥ ''x'' # ''x''CC = ''x''C # (''x'' + ''y'')C = ''x''C + ''y' ... [...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] |
Closed Set
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a closed set is a set which is closed under the limit operation. This should not be confused with a closed manifold. Equivalent definitions By definition, a subset A of a topological space (X, \tau) is called if its complement X \setminus A is an open subset of (X, \tau); that is, if X \setminus A \in \tau. A set is closed in X if and only if it is equal to its closure in X. Equivalently, a set is closed if and only if it contains all of its limit points. Yet another equivalent definition is that a set is closed if and only if it contains all of its boundary points. Every subset A \subseteq X is always contained in its (topological) closure in X, which is denoted by \operatorname_X A; that is, if A \subseteq X then A \subseteq \oper ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
