Free products of unital ℓ-groups and free products of generalized MV-algebras

Dvurečenskij, Anatolij; Holland, Walter W.

doi:10.1007/s00012-010-0035-x

Cited by 6 publications

(10 citation statements)

References 17 publications

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“…The main result in Dvurečenskij and Holland (2009c) says that the free product of any collection of non-trivial GMV-algebras exists in the variety of all GMV-algebras. If we restrict to particular families of GMV-algebras, it is possible to show that sometimes if the free product exists in a smaller family, so exists in a bigger one and both coincide (Dvurečenskij and Holland 2009c, Lem 4.2):…”

Section: Applications Of Cyclic Elementsmentioning

confidence: 99%

“…The free product of MV-algebras was studied in Mundici (1988) and the free product of GMV-algebras was exhibit in Dvurečenskij and Holland (2009c): let V be a class of GMV-algebras and let fM t g t2T be a family of GMV-algebras in V: A V-free product (or simply a free product if V is known from the context) of this family is a GMV-algebra M 2 V; denoted by F V t2T M t ; together with a family of GMV-embeddings ff t : M t ! Mg t2T such that…”

Section: Applications Of Cyclic Elementsmentioning

confidence: 99%

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Cyclic elements and subalgebras of GMV-algebras

Dvurečenskij

2009

Soft Comput

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We study cyclic elements of order n of GMValgebras. The existence of cyclic elements is equivalent to the condition that a given GMV-algebra contains a copy of the MV-algebra CðZ; nÞ: Using cyclic elements, we describe necessary conditions which guarantee the existence of a greatest subalgebra belonging to the variety generated by CðZ; nÞ: This is true, e.g. for representable GMV-algebras. Finally, we use cyclic elements to prove the existence of a free product in various varieties of GMValgebras.

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Section: Applications Of Cyclic Elementsmentioning

confidence: 99%

Section: Applications Of Cyclic Elementsmentioning

confidence: 99%

Cyclic elements and subalgebras of GMV-algebras

Dvurečenskij

2009

Soft Comput

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“…an element u ∈ G + such that given g ∈ G, there is an integer n ≥ 1 such that g ≤ nu). Using this basic representation, a free product of GMV-algebras was studied in [5] and the free product of MV-algebras in [13]. In [5], it was shown that the free product of MV-algebras in the category of GMV-algebras can be non-commutative.…”

Section: Introductionmentioning

confidence: 98%

“…Using this basic representation, a free product of GMV-algebras was studied in [5] and the free product of MV-algebras in [13]. In [5], it was shown that the free product of MV-algebras in the category of GMV-algebras can be non-commutative. In what follows, using the bounded Boolean power, we show that the free product of an MV-algebra with a Boolean algebra 2 n taken in the category of GMValgebras is again an MV-algebra.…”

Section: Introductionmentioning

confidence: 98%

Bounded Boolean Powers and Free Product of GMV-Algebras

Dvurečenskij

Hyčko

2010

Int J Theor Phys

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GMV-algebras are a non-commutative generalization of MV-algebras and they describe a non-commutative many-valued Łukasiewicz logic. We introduce a bounded Boolean power of GMV-algebras. Using the bounded Boolean power of MV-algebras, we show that the free product of an MV-algebra and a Boolean algebra is again a commutative MV-algebra, however the free product of MV-algebras in the variety of GMValgebras is in general non-commutative (Dvurečenskij and Holland, Algebra Univers., (2010), doi:10.1007/S00012-010-0035-X). In addition, we analyze also a topological version of the Boolean power and we show that these constructions are equivalent.

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“…This provided a way to study varieties of pseudo MV-algebras by examining equational classes of "unital" lattice-ordered groups. Holland has been at the forefront of this research ever since and obtained major results in the last few years, with A. Dvurečenskij, M. R. Darnel and M. Droste [59,63,64,65,66,67,70,71,72].…”

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confidence: 99%

W. Charles Holland, 75th birthday

Ball

Glass

Martínez

et al. 2011

Mathematica Slovaca

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ABSTRACT. This issue of Mathematica Slovaca is in honour of W. Charles Holland's 75th birthday. We present here a brief account of some of his research (to date) and a couple of brief personal sketches of the man. Research of W. Charles HollandIn the beginning were (totally) ordered groups: groups with a total order preserved on both sides by the group operation. Then came the generalisation to lattice-ordered groups; a lattice-ordered group is a group equipped with a lattice order, with the order preserved on both sides by the group operation. For any totally ordered set Ω, the group of order-preserving permutations (automorphisms) Aut(Ω, ≤) of (Ω, ≤) forms a lattice-ordered group under the pointwise order. A lattice-ordered group arising in this way is called a lattice-ordered permutation group (or -permutation group). The main role of -permutation groups was to provide examples of lattice-ordered groups. In 1963, Holland changed the playing field by showing that every lattice-ordered group is ( -isomorphic to) a sublattice subgroup of some -permutation group! Lattice-ordered-group results could now be obtained by working instead with -permutation groups, which are far more concrete objects, amenable to drawing pictures. This great breakthrough is examined in depth in [BDG].Totally ordered groups were studied in the early part of the twentieth century. Paul F. Conrad, Holland's Ph.D. Dissertation Director, took this up but viewed totally ordered groups as a subclass of the class lattice-ordered groups. This permitted purely algebraic methods which Conrad exploited to great effect.

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Free products of unital ℓ-groups and free products of generalized MV-algebras

Abstract: We show that the free product of any collection of non-trivial unitalgroups with fixed strong unit exists. Equivalently, the free product of any collection of non-trivial generalized MV-algebras exists. We then investigate free products in some of the varieties of these algebras.

Cited by 6 publications

References 17 publications

Cyclic elements and subalgebras of GMV-algebras

Cyclic elements and subalgebras of GMV-algebras

Bounded Boolean Powers and Free Product of GMV-Algebras

W. Charles Holland, 75th birthday

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