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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 group (mathematics), groups, ring (mathematics), rings, field (mathematics), fields, module (mathematics), modules, vector spaces, lattice (order), lattices, and algebra over a field, 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 variable (mathematics), variables to represent numbers in computation and reasoning. Algebraic structures, with their associated homomorphisms, form category (mathematics), 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 exampl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
