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MV-algebras
In abstract algebra, a branch of pure mathematics, an MV-algebra is an algebraic structure with a binary operation \oplus, a unary operation \neg, and the constant 0, satisfying certain axioms. MV-algebras are the algebraic semantics of ﾅ「kasiewicz logic; the letters MV refer to the ''many-valued'' logic of ﾅ「kasiewicz. MV-algebras coincide with the class of bounded commutative BCK algebras. Definitions An MV-algebra is an algebraic structure \langle A, \oplus, \lnot, 0\rangle, consisting of * a non-empty set A, * a binary operation \oplus on A, * a unary operation \lnot on A, and * a constant 0 denoting a fixed element of A, which satisfies the following identities: * (x \oplus y) \oplus z = x \oplus (y \oplus z), * x \oplus 0 = x, * x \oplus y = y \oplus x, * \lnot \lnot x = x, * x \oplus \lnot 0 = \lnot 0, and * \lnot ( \lnot x \oplus y)\oplus y = \lnot ( \lnot y \oplus x) \oplus x. By virtue of the first three axioms, \langle A, \oplus, 0 \rangle is a commutativ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
ﾅ「kasiewicz Logic
In mathematics and philosophy, ﾅ「kasiewicz logic ( , ) is a non-classical, many-valued logic. It was originally defined in the early 20th century by Jan ﾅ「kasiewicz as a three-valued logic;ﾅ「kasiewicz J., 1920, O logice trﾃｳjwartoﾅ嫩iowej (in Polish). Ruch filozoficzny 5:170窶171. English translation: On three-valued logic, in L. Borkowski (ed.), ''Selected works by Jan ﾅ「kasiewicz'', North窶滴olland, Amsterdam, 1970, pp. 87窶88. it was later generalized to ''n''-valued (for all finite ''n'') as well as infinitely-many-valued ( 邃ｵ0-valued) variants, both propositional and first-order.Hay, L.S., 1963Axiomatization of the infinite-valued predicate calculus ''Journal of Symbolic Logic'' 28:77窶86. The 邃ｵ0-valued version was published in 1930 by ﾅ「kasiewicz and Alfred Tarski; consequently it is sometimes called the ﾅ「kasiewiczTarski logic. citing ﾅ「kasiewicz, J., Tarski, A.Untersuchungen ﾃｼber den Aussagenkalkﾃｼl Comp. Rend. Soc. Sci. et Lettres Varsovie Cl. III 23, 30窶5 ... [...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, JarosﾅＢw (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] |
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] |
BL (logic)
In mathematical logic, basic fuzzy logic (or shortly BL), the logic of the continuous t-norms, is one of the t-norm fuzzy logics. It belongs to the broader class of substructural logics, or logics of residuated lattices;Ono (2003). it extends the logic MTL of all left-continuous t-norms. Syntax Language The language of the propositional logic BL consists of countably many propositional variables and the following primitive logical connectives: * Implication \rightarrow (binary) * Strong conjunction \otimes (binary). The sign & is a more traditional notation for strong conjunction in the literature on fuzzy logic, while the notation \otimes follows the tradition of substructural logics. * Bottom \bot (nullary 窶 a propositional constant); 0 or \overline are common alternative signs and zero a common alternative name for the propositional constant (as the constants bottom and zero of substructural logics coincide in MTL). The following are the most common defined logical ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Many-valued Logic
Many-valued logic (also multi- or multiple-valued logic) refers to a propositional calculus in which there are more than two truth values. Traditionally, in Aristotle's logical calculus, there were only two possible values (i.e., "true" and "false") for any proposition. Classical two-valued logic may be extended to ''n''-valued logic for ''n'' greater than 2. Those most popular in the literature are three-valued (e.g., ﾅ「kasiewicz's and Kleene's, which accept the values "true", "false", and "unknown"), four-valued, nine-valued, the finite-valued (finitely-many valued) with more than three values, and the infinite-valued (infinitely-many-valued), such as fuzzy logic and probability logic. History It is wrong that the first known classical logician who did not fully accept the law of excluded middle was Aristotle (who, ironically, is also generally considered to be the first classical logician and the "father of wo-valuedlogic"). In fact, Aristotle did not contest the univer ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Approximately Finite-dimensional C*-algebra
In mathematics, an approximately finite-dimensional (AF) C*-algebra is a C*-algebra that is the inductive limit of a sequence of finite-dimensional C*-algebras. Approximate finite-dimensionality was first defined and described combinatorially by Ola Bratteli. Later, George A. Elliott gave a complete classification of AF algebras using the ''K''0 functor whose range consists of ordered abelian groups with sufficiently nice order structure. The classification theorem for AF-algebras serves as a prototype for classification results for larger classes of separable simple amenable stably finite C*-algebras. Its proof divides into two parts. The invariant here is ''K''0 with its natural order structure; this is a functor. First, one proves ''existence'': a homomorphism between invariants must lift to a *-homomorphism of algebras. Second, one shows ''uniqueness'': the lift must be unique up to approximate unitary equivalence. Classification then follows from what is known as ''the inte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Categorical Equivalence
In category theory, a branch of abstract mathematics, an equivalence of categories is a relation between two categories that establishes that these categories are "essentially the same". There are numerous examples of categorical equivalences from many areas of mathematics. Establishing an equivalence involves demonstrating strong similarities between the mathematical structures concerned. In some cases, these structures may appear to be unrelated at a superficial or intuitive level, making the notion fairly powerful: it creates the opportunity to "translate" theorems between different kinds of mathematical structures, knowing that the essential meaning of those theorems is preserved under the translation. If a category is equivalent to the opposite (or dual) of another category then one speaks of a duality of categories, and says that the two categories are dually equivalent. An equivalence of categories consists of a functor between the involved categories, which is required t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lattice-ordered Group
In abstract algebra, a partially ordered group is a group (''G'', +) equipped with a partial order "竕､" that is ''translation-invariant''; in other words, "竕､" has the property that, for all ''a'', ''b'', and ''g'' in ''G'', if ''a'' 竕､ ''b'' then ''a'' + ''g'' 竕､ ''b'' + ''g'' and ''g'' +'' a'' 竕､ ''g'' +'' b''. An element ''x'' of ''G'' is called positive if 0 竕､ ''x''. The set of elements 0 竕､ ''x'' is often denoted with ''G''+, and is called the positive cone of ''G''. By translation invariance, we have ''a'' 竕､ ''b'' if and only if 0 竕､ -''a'' + ''b''. So we can reduce the partial order to a monadic property: if and only if For the general group ''G'', the existence of a positive cone specifies an order on ''G''. A group ''G'' is a partially orderable group if and only if there exists a subset ''H'' (which is ''G''+) of ''G'' such that: * 0 竏 ''H'' * if ''a'' 竏 ''H'' and ''b'' 竏 ''H'' then ''a'' + ''b'' 竏 ''H'' * if ''a'' 竏 ''H'' then -''x'' + ''a'' + '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linearly Ordered Group
In mathematics, specifically abstract algebra, a linearly ordered or totally ordered group is a group ''G'' equipped with a total order "竕､" that is ''translation-invariant''. This may have different meanings. We say that (''G'', 竕､) is a: * left-ordered group if 竕､ is left-invariant, that is ''a'' 竕､ ''b'' implies ''ca'' 竕､ ''cb'' for all ''a'', ''b'', ''c'' in ''G'', * right-ordered group if 竕､ is right-invariant, that is ''a'' 竕､ ''b'' implies ''ac'' 竕､ ''bc'' for all ''a'', ''b'', ''c'' in ''G'', * bi-ordered group if 竕､ is bi-invariant, that is it is both left- and right-invariant. A group ''G'' is said to be left-orderable (or right-orderable, or bi-orderable) if there exists a left- (or right-, or bi-) invariant order on ''G''. A simple necessary condition for a group to be left-orderable is to have no elements of finite order; however this is not a sufficient condition. It is equivalent for a group to be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Order Type
In mathematics, especially in set theory, two ordered sets and are said to have the same order type if they are order isomorphic, that is, if there exists a bijection (each element pairs with exactly one in the other set) f\colon X \to Y such that both and its inverse are monotonic (preserving orders of elements). In the special case when is totally ordered, monotonicity of implies monotonicity of its inverse. For example, the set of integers and the set of even integers have the same order type, because the mapping n\mapsto 2n is a bijection that preserves the order. But the set of integers and the set of rational numbers (with the standard ordering) do not have the same order type, because even though the sets are of the same size (they are both countably infinite), there is no order-preserving bijective mapping between them. To these two order types we may add two more: the set of positive integers (which has a least element), and that of negative integers (which has a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Infinitesimal
In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally referred to the " infinity- th" item in a sequence. Infinitesimals do not exist in the standard real number system, but they do exist in other number systems, such as the surreal number system and the hyperreal number system, which can be thought of as the real numbers augmented with both infinitesimal and infinite quantities; the augmentations are the reciprocals of one another. Infinitesimal numbers were introduced in the development of calculus, in which the derivative was first conceived as a ratio of two infinitesimal quantities. This definition was not rigorously formalized. As calculus developed further, infinitesimals were replaced by limits, which can be calculated using the standard real numbers. Infinitesimals regained popularit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Two-element Boolean Algebra
In mathematics and abstract algebra, the two-element Boolean algebra is the Boolean algebra whose ''underlying set'' (or universe or ''carrier'') ''B'' is the Boolean domain. The elements of the Boolean domain are 1 and 0 by convention, so that ''B'' = . Paul Halmos's name for this algebra "2" has some following in the literature, and will be employed here. Definition ''B'' is a partially ordered set and the elements of ''B'' are also its bounds. An operation of arity ''n'' is a mapping from ''B''n to ''B''. Boolean algebra consists of two binary operations and unary complementation. The binary operations have been named and notated in various ways. Here they are called 'sum' and 'product', and notated by infix '+' and '竏', respectively. Sum and product commute and associate, as in the usual algebra of real numbers. As for the order of operations, brackets are decisive if present. Otherwise '竏' precedes '+'. Hence ''A竏傳 + C'' is parsed as ''(A竏傳)& ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
