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Łukasiewicz–Moisil Algebra
Łukasiewicz–Moisil algebras (LM''n'' algebras) were introduced in the 1940s by Grigore Moisil (initially under the name of Łukasiewicz algebras) in the hope of giving algebraic semantics for the ''n''-valued Łukasiewicz logic. However, in 1956 Alan Rose discovered that for ''n'' ≥ 5, the Łukasiewicz–Moisil algebra does not model the Łukasiewicz logic. A faithful model for the ℵ0-valued (infinitely-many-valued) Łukasiewicz–Tarski logic was provided by C. C. Chang's MV-algebra, introduced in 1958. For the axiomatically more complicated (finite) ''n''-valued Łukasiewicz logics, suitable algebras were published in 1977 by Revaz Grigolia and called MV''n''-algebras. MV''n''-algebras are a subclass of LM''n''-algebras, and the inclusion is strict for ''n'' ≥ 5.Iorgulescu, A.: Connections between MV''n''-algebras and ''n''-valued Łukasiewicz-Moisil algebras—I. Discrete Math. 181, 155–177 (1998) In 1982 Roberto Cignoli published some additional constraints that a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grigore Moisil
Grigore Constantin Moisil (; 10 January 1906 – 21 May 1973) was a Romanian mathematician, computer pioneer, and list of members of the Romanian Academy, titular member of the Romanian Academy. His research was mainly in the fields of mathematical logic (Łukasiewicz–Moisil algebra), algebraic logic, MV-algebra, and differential equations. He is viewed as the father of computer science in Romania. Moisil was also a member of the Academy of Sciences of Bologna and of the International Institute of Philosophy. In 1996, the IEEE Computer Society awarded him posthumously the ''Computer Pioneer'' Award. Biography Grigore Moisil was born in 1906 in Tulcea into an intellectual family. His great-grandfather, Grigore Moisil (1814–1891), a clergyman, was one of the founders of the George Coșbuc National College (Năsăud), first Romanian high school in Năsăud. His father, Constantin Moisil (1876–1958), was a history professor, archaeology, archaeologist and numismatics, numismat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
De Morgan Algebra
__NOTOC__ In mathematics, a De Morgan algebra (named after Augustus De Morgan, a British mathematician and logician) is a structure ''A'' = (A, ∨, ∧, 0, 1, ¬) such that: * (''A'', ∨, ∧, 0, 1) is a bounded distributive lattice, and * ¬ is a De Morgan involution: ¬(''x'' ∧ ''y'') = ¬''x'' ∨ ¬''y'' and ¬¬''x'' = ''x''. (i.e. an involution that additionally satisfies De Morgan's laws) In a De Morgan algebra, the laws * ¬''x'' ∨ ''x'' = 1 (law of the excluded middle), and * ¬''x'' ∧ ''x'' = 0 (law of noncontradiction) do not always hold. In the presence of the De Morgan laws, either law implies the other, and an algebra which satisfies them becomes a Boolean algebra. Remark: It follows that ¬(x ∨ y) = ¬x ∧ ¬y, ¬1 = 0 and ¬0 = 1 (e.g. ¬1 = ¬1 ∨ 0 = ¬1 ∨ ¬¬0 = ¬(1 ∧ ¬0) = ¬¬0 = 0). Thus ¬ is a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monadic Boolean Algebra
In abstract algebra, a monadic Boolean algebra is an algebraic structure ''A'' with signature :⟨·, +, ', 0, 1, ∃⟩ of type ⟨2,2,1,0,0,1⟩, where ⟨''A'', ·, +, ', 0, 1⟩ is a Boolean algebra. The monadic/unary operator ∃ denotes the existential quantifier, which satisfies the identities (using the received prefix notation for ∃): * ∃0 = 0 * ∃''x'' ≥ ''x'' * ∃(''x'' + ''y'') = ∃''x'' + ∃''y'' * ∃''x''∃''y'' = ∃(''x''∃''y''). ∃''x'' is the ''existential closure'' of ''x''. Dual to ∃ is the unary operator ∀, the universal quantifier, defined as ∀''x'' := (∃''x' '')'. A monadic Boolean algebra has a dual definition and notation that take ∀ as primitive and ∃ as defined, so that ∃''x'' := (∀''x'' ' )' . (Compare this with the definition of the dual Boolean algebra.) Hence, with this notation, an algebra ''A'' has signature ⟨·, +, ', 0, 1, ∀⟩, with ⟨''A'', ·, +, ', 0, 1& ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Antonio Monteiro (mathematician)
António Aniceto Monteiro (1907–1980) was a mathematician born in Portuguese Angola who later emigrated to Brazil in 1945 and finally to Argentina in 1950. Monteiro is best known for establishing a school of algebraic logic at Universidad Nacional del Sur, Bahía Blanca, Argentina. His efforts to promote theoretical computer science research in Argentina were less successful.http://publicaciones.dc.uba.ar/Publications/2012/CR12a/MonteiroPionero.pdf After his undergraduate studies at the University of Lisbon (completed in 1930), Monteiro obtained a PhD at Sorbonne in 1936 under the advisement of Maurice Fréchet with a thesis in topology. In Portugal Monteiro was the main founder of the journal '' Portugaliae Mathematica'' in 1937. In 1945 Monteiro moved to Brazil. There are two versions of why Monteiro left Portugal. The first version is that Monteiro and other Portuguese mathematicians like Ruy Luís Gomes fell foul of Salazar's regime for their political beliefs; some, like ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Substructure (mathematics)
In mathematical logic, an (induced) substructure or (induced) subalgebra is a structure whose domain is a subset of that of a bigger structure, and whose functions and relations are restricted to the substructure's domain. Some examples of subalgebras are subgroups, submonoids, subrings, subfields, subalgebras of algebras over a field, or induced subgraphs. Shifting the point of view, the larger structure is called an extension or a superstructure of its substructure. In model theory, the term "submodel" is often used as a synonym for substructure, especially when the context suggests a theory of which both structures are models. In the presence of relations (i.e. for structures such as ordered groups or graphs, whose signature is not functional) it may make sense to relax the conditions on a subalgebra so that the relations on a weak substructure (or weak subalgebra) are ''at most'' those induced from the bigger structure. Subgraphs are an example where the distinction mat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subdirect Product
In mathematics, especially in the areas of abstract algebra known as universal algebra, group theory, ring theory, and module theory, a subdirect product is a subalgebra of a direct product that depends fully on all its factors without however necessarily being the whole direct product. The notion was introduced by Garrett Birkhoff, Birkhoff in 1944 and has proved to be a powerful generalization of the notion of direct product. Definition A subdirect product is a subalgebra (in the sense of 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 ...) ''A'' of a direct product Π''iAi'' such that every induced projection (the composite ''pjs'': ''A'' → ''Aj'' of a projection ''p''''j'': Π''iAi'' → ''Aj'' with the subalgebra inclusion ''s'': ''A'' → Π''iAi'') is su ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Direct Product
In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one talks about the product in category theory, which formalizes these notions. Examples are the product of sets, groups (described below), rings, and other algebraic structures. The product of topological spaces is another instance. There is also the direct sum – in some areas this is used interchangeably, while in others it is a different concept. Examples * If we think of \R as the set of real numbers, then the direct product \R \times \R is just the Cartesian product \. * If we think of \R as the group of real numbers under addition, then the direct product \R\times \R still has \ as its underlying set. The difference between this and the preceding example is that \R \times \R is now a group, and so we have to also say how to add their ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Embedding
In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup. When some object X is said to be embedded in another object Y, the embedding is given by some injective and structure-preserving map f:X\rightarrow Y. The precise meaning of "structure-preserving" depends on the kind of mathematical structure of which X and Y are instances. In the terminology of category theory, a structure-preserving map is called a morphism. The fact that a map f:X\rightarrow Y is an embedding is often indicated by the use of a "hooked arrow" (); thus: f : X \hookrightarrow Y. (On the other hand, this notation is sometimes reserved for inclusion maps.) Given X and Y, several different embeddings of X in Y may be possible. In many cases of interest there is a standard (or "canonical") embedding, like those of the natural numbers in the integers, the integers in the rational numbers, the rational n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pointwise
In mathematics, the qualifier pointwise is used to indicate that a certain property is defined by considering each value f(x) of some function f. An important class of pointwise concepts are the ''pointwise operations'', that is, operations defined on functions by applying the operations to function values separately for each point in the domain of definition. Important relations can also be defined pointwise. Pointwise operations Formal definition A binary operation on a set can be lifted pointwise to an operation on the set of all functions from to as follows: Given two functions and , define the function by Commonly, ''o'' and ''O'' are denoted by the same symbol. A similar definition is used for unary operations ''o'', and for operations of other arity. Examples \begin (f+g)(x) & = f(x)+g(x) & \text \\ (f\cdot g)(x) & = f(x) \cdot g(x) & \text \\ (\lambda \cdot f)(x) & = \lambda \cdot f(x) & \text \end where f, g : X \to R. See also pointwise product, and scalar. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boolean Algebra (structure)
In abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice. This type of algebraic structure captures essential properties of both set operations and logic operations. A Boolean algebra can be seen as a generalization of a power set algebra or a field of sets, or its elements can be viewed as generalized truth values. It is also a special case of a De Morgan algebra and a Kleene algebra (with involution). Every Boolean algebra gives rise to a Boolean ring, and vice versa, with ring multiplication corresponding to conjunction or meet ∧, and ring addition to exclusive disjunction or symmetric difference (not disjunction ∨). However, the theory of Boolean rings has an inherent asymmetry between the two operators, while the axioms and theorems of Boolean algebra express the symmetry of the theory described by the duality principle. __TOC__ History The term "Boolean algebra" honors George Boole (1815–1864), a self-educated English ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Endomorphism
In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space is a linear map , and an endomorphism of a group is a group homomorphism . In general, we can talk about endomorphisms in any category. In the category of sets, endomorphisms are functions from a set ''S'' to itself. In any category, the composition of any two endomorphisms of is again an endomorphism of . It follows that the set of all endomorphisms of forms a monoid, the full transformation monoid, and denoted (or to emphasize the category ). Automorphisms An invertible endomorphism of is called an automorphism. The set of all automorphisms is a subset of with a group structure, called the automorphism group of and denoted . In the following diagram, the arrows denote implication: Endomorphism rings Any two endomorphisms of an abelian group, , can be added toge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modal Logic
Modal logic is a collection of formal systems developed to represent statements about necessity and possibility. It plays a major role in philosophy of language, epistemology, metaphysics, and natural language semantics. Modal logics extend other systems by adding unary operators \Diamond and \Box, representing possibility and necessity respectively. For instance the modal formula \Diamond P can be read as "possibly P" while \Box P can be read as "necessarily P". Modal logics can be used to represent different phenomena depending on what kind of necessity and possibility is under consideration. When \Box is used to represent epistemic necessity, \Box P states that P is epistemically necessary, or in other words that it is known. When \Box is used to represent deontic necessity, \Box P states that P is a moral or legal obligation. In the standard relational semantics for modal logic, formulas are assigned truth values relative to a ''possible world''. A formula's truth value at ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
