Partially Ordered Groups

Glass, A. M. W.

doi:10.1142/3811

Cited by 188 publications

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“…This variety provides algebraic semantics for the full Lambek calculus (pointed residuated lattices are therefore often referred to also as FL-algebras) and its subvarieties correspond to substructural logics. Moreover, lattice-ordered groups (or -groups) (see [3], [26]) can be presented as residuated lattices satisfying x(x\e) = e. It suffices to let x\y = x −1 y and y/x = yx −1 .…”

Section: Residuated Latticesmentioning

confidence: 99%

Amalgamation and interpolation in ordered algebras

2014

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The first part of this paper provides a comprehensive and self-contained account of the interrelationships between algebraic properties of varieties and properties of their free algebras and equational consequence relations. In particular, proofs are given of known equivalences between the amalgamation property and the Robinson property, the congruence extension property and the extension property, and the flat amalgamation property and the deductive interpolation property, as well as various dependencies between these properties. These relationships are then exploited in the second part of the paper in order to provide new proofs of amalgamation and deductive interpolation for the varieties of lattice-ordered abelian groups and MV-algebras, and to determine important subvarieties of residuated lattices where these properties hold or fail. In particular, a full description is given of all subvarieties of commutative GMV-algebras possessing the amalgamation property.

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Section: Residuated Latticesmentioning

confidence: 99%

Amalgamation and interpolation in ordered algebras

2014

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“…Then (W ; * , −1 , (0, e A ), ≤) is a po-group called the (unrestricted) Wreath product of A and G; we write also W = A Wr G, see [Gla,Ex 1.3.27]. If G is an ℓ-group, so is the Wreath product.…”

Section: Wreath Product and Rdp'smentioning

confidence: 99%

“…Besides these restricted Wreath products we introduce according to [Gla,Ex 1.3.28] two its special kinds: The subgroup of W = Z⋉G Z consisting of all elements of the form (n, g i : i ∈ Z ), where g i = e for all but finitely many i ∈ Z, can be ordered in two ways: an element (n, g i : i ∈ Z ) ∈ W is positive if either n > 0 or n = 0 and g j > e where j is the greatest integer i such that g i = e. We call W the right wreath product of Z and G, and we writeW = Z − → wr G. The second ordering of W is as follows: An element (n, g i : i ∈ Z ) ∈ W is positive iff either n > 0 or n = 0 and g j > e where j is the least integer i such that g i = e. We call W the left wreath product of Z, and we write W = Z ← − wr G. Both products are po-groups, and if G is a linearly ordered group so are both ones. We note that according to [Dar,Cor 61.17], in such a case both products generate different varieties of ℓ-groups.…”

Section: Wreath Product and Rdp'smentioning

confidence: 99%

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Riesz decomposition properties and the lexicographic product of po-groups

Dvurečenskij

2015

Soft Comput

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Abstract. We establish conditions when a certain type of the Riesz Decomposition Property (RDP) holds in the lexicographic product of two po-groups. It is well known that the resulting product is an ℓ-group if and only if the first one is linearly ordered and the second one is an ℓ-group. This can be equivalently studied as po-groups with a special type of the RDP. In the paper we study three different types of RDP's. RDP's of the lexicographic products are important for the study of pseudo effect algebras where infinitesimal elements play an important role both for algebras as well as for the first order logic of valid but not provable formulas.

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“…In fact, if M = Γ (G, u), where (G, u) is a unital ℓ-group, then by [6], (M ; ∨, ∧) is a complete lattice if and only if G is a complete ℓ-group. In [9]), therefore every non-commutative GM V -algebra is not complete. Hence we will consider any GM V -subalgebra A of the GM V -algebra M X satisfying following conditions:…”

Section: Functional Monadic Gm V -Algebrasmentioning

confidence: 99%

Monadic GMV-algebras

Rachůnek

Šalounová

2008

Arch. Math. Logic

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Monadic M V -algebras are an algebraic model of the predicate calculus of the Lukasiewicz infinite valued logic in which only a single individual variable occurs. GM V -algebras are a non-commutative generalization of M V -algebras and are an algebraic counterpart of the non-commutative Lukasiewicz infinite valued logic. We introduce monadic GM V -algebras and describe their connections to certain couples of GM V -algebras and to left adjoint mappings of canonical embeddings of GM Valgebras. Furthermore, functional M GM V -algebras are studied and polyadic GM Valgebras are introduced and discussed.

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Partially Ordered Groups

Cited by 188 publications

References 0 publications

Amalgamation and interpolation in ordered algebras

Amalgamation and interpolation in ordered algebras

Riesz decomposition properties and the lexicographic product of po-groups

Monadic GMV-algebras

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