D-lattices

Chovanec, Ferdinand; K�pka, Franti?ek

doi:10.1007/bf00676241

Cited by 22 publications

(16 citation statements)

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“…Alternatively, an MV-effect algebra may be defined as a lattice ordered effect algebra with the property ∀a, b ∈ E : (a ∨ b) a = b (a ∧ b) (Chovanec and Kôpka 1995;Dvurečenskij and Pulmannová 2000). It was proved in Chovanec and Kôpka (1995) that MV-effect algebras are in one-to-one correspondence with MV-algebras.…”

Section: Preliminariesmentioning

confidence: 99%

“…It was proved in Chovanec and Kôpka (1995) that MV-effect algebras are in one-to-one correspondence with MV-algebras. Boolean algebras coincide with MV-effect algebras with the property ∀a ∈ E : a ∧ a = 0.…”

Section: Preliminariesmentioning

confidence: 99%

“…It turned out that MV-algebras are a subclass of a more general class of effect algebras (Chovanec and Kôpka 1995). Effect algebras were introduced, for the purposes of mathematical foundations of quantum theories, as an abstract algebraic generalization of the structure of Hilbert space effects, that is, self-adjoint operators between the zero and identity operator (with respect to the usual ordering) on a Hilbert space (Foulis and Bennett 1994;Kôpka and Chovanec 1994;Dvurečenskij and Pulmannová 2000).…”

Section: Introductionmentioning

confidence: 99%

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Ideals in MV-pairs

Pulmannová

Vinceková

2008

Soft Comput

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Section: Preliminariesmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Ideals in MV-pairs

Pulmannová

Vinceková

2008

Soft Comput

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“…A special case of a commutative minimal clan is an MV-algebra introduced by Chang [5] in 1957, where MV is supposed to suggest many-valued logic (see also [25], [27], [6] for alternative definitions of MV-algebras).…”

Section: Introductionmentioning

confidence: 99%

“…For a generalization of test spaces where the logic becomes a D-poset, see [8], [9], [37], [32]. In [6], there is shown that every MV-algebra is a D-poset (a D-lattice, more precisely), and conditions under which a D-lattice becomes an MV-algebra are given. For more details about relations among commutative minimal clans, MV-algebras and D-posets see [31].…”

Section: Introductionmentioning

confidence: 99%

Congruences in partial abelian semigroups

Pulmannová

1997

Algebra Universalis

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A partial abelian semigroup (PAS) is a structure (L, Þ, ), where is a partial binary operation on L with domain Þ, which is commutative and associative (whenever the corresponding elements exist). A class of congruences on partial abelian semigroups are studied such that the corresponding quotient is again a PAS. If M is a subset of a PAS L, we say that x, y L are perspective with respect to M, if there is z L such that x z M and y z M. A subset M is called weakly algebraic if perspectivity with respect to M is a congruence. Some conditions are shown under which a congruence coincides with a perspectivity with respect to an appropriate set M. Especially, conditions under which the corresponding quotient is a D-poset are found. It is also shown that every congruence of MV-algebras and orthomodular lattices is given by a perspectivity with respect to an appropriate set M. IntroductionIn this paper, we will study some congruences on partial abelian semigroups, where a partial abelian semigroup is a structure (L, Þ, ), where is a partial binary operation on L with domain Þ, which is commutative and associative in a restricted sense ([37]). Similar structures have already appeared in the literature and found applications in several fields.A general theory of universal partial algebras can be found in [3]. A special case of partial groupoids -algebras with a partially defined binary operation -have been studied in [24].A common abstraction of Boolean rings and commutative lattice-ordered groups was introduced by Schmidt in 1985 [35]. In [36] such a not necessarily commutative partial semigroup structure is studied under the name minimal clan. A commutative minimal clan is a set E with a set S of pairs of summable elements of E, a partial addition +: S E, and an order relation 5 such that (E, S, +, 5) is a commutative lattice-ordered partial semigroup having the cancellative property Presented by R. W. Quackenbush. 1991 AMS Subject Classification: 06A06, 08A55, 06C15. Key words and phrases. Partial abelian semigroup, effect algebra, difference (operation) on a poset, orthomodular poset, orthoalgebra, difference poset, homomorphism and congruence of partial abelian semigroups, algebraic set. 119

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MV-pairs and states

Pulmannová

2008

Soft Comput

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An MV-pair is a BG-pair (B; G) (where B is a Boolean algebra and G is a subgroup of the automorphism group of B) satisfying certain conditions. Recently, it was proved by Jenca that, given an MV-pair (B; G), the quotient ðB=$ G Þ; where ð $ G Þ; is an equivalence relation naturally associated with G, is an MV-algebra, and conversely, to every MV-algebra there corresponds an MV-pair. In this paper, we introduce a new definition of so called MV*-pair, and we show that ðB=$ G Þ; is an effect algebra iff the first of the defining properties of the MV*-pair is satisfied, while the second property guarantees the MV-algebra structure of ðB=$ G Þ. We also study some relations between states on MV-algebras and the corresponding R-generated Boolean algebras.Keywords MV-algebra Á Effect algebra Á R-generated Boolean algebra Á MV-pair Á State 1 MV-algebras and MV-effect algebras MV-algebras were introduced by Chang in 1958 (Chang 1959) as a model for many-valued logics. The original Chang's definition was simplified by Mangani in 1973. An MV-algebra ðM; È; Ã ; 0Þ is a (2, 1, 0) type of algebra, where the binary operation È is commutative and associative with the nullary operation 0 as neutral element;The last of the axioms is called Łukasiewicz axiom and it is the one which guarantees that MV-algebra is a lattice which is moreover distributive, according to the ordering given by a b iff a Ã È b ¼ 1: Moreover, in 1986, a famous representation theorem by Mundici furnishes the concept of MV-algebra by isomorphism between MValgebras and intervals [0, u] in Abelian lattice-ordered groups with strong unit u. We have a prototypical model of MV-algebra in the real unit interval [0, 1].It turns out that MV-algebras are a subclass of a more general algebraic structures called effect algebras. An effect algebra is a partial algebraic structure ðE; þ; 0; 1Þ with a partial binary operation ? satisfying (1) a þ b ¼ b þ a; (2) ða þ bÞ þ c ¼ a þ ðb þ cÞ (in the sense that if one side exists so does the other and equality holds); (3) to every a 2 E; there is a unique element a 0 such that a þ a 0 ¼ 1; (4) if a þ 1 is defined then a ¼ 0: We say that a and b are orthogonal (written a ? b) if a þ b is defined. Sometimes we write a þ b tacitly assuming that a ? b: Notice that the element a 1 þ a 2 þ Á Á Á þ a n ¼: P n i¼1 a i can be defined recurrently if all the partial sub-sums exist. An effect algebra is partially ordered by the relation a b if there is c 2 E with a þ c ¼ b: If such an element c exists, it is unique, and is denoted by c ¼ b À a: In the latter ordering, 0 is the smallest and 1 is the greatest element. Effect algebras were introduced in Foulis and Bennett (1994), in an alternative form called D-posets in Kôpka and Chovanec (1994).We may define a partial operation ? on an MV-algebra by the restriction of È to the pairs of elements for which a b Ã : Then the structure ðM; þ; 0; 1Þ is a lattice ordered effect algebra that satisfies some additional conditions. Also a reversed process can be done to obtain an

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D-lattices

Cited by 22 publications

References 5 publications

Ideals in MV-pairs

Ideals in MV-pairs

Congruences in partial abelian semigroups

MV-pairs and states

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