Subdirectly irreducible generalized sums of upper semilattice ordered systems of algebras

Płonka, J.; Szylicka, Z.

doi:10.1007/s00012-002-8184-1

Cited by 3 publications

(2 citation statements)

References 8 publications

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“…All these results point toward the idea that locally integral ipo-semigroups are built up from integral ipo-monoids, or at least their semigroup reducts are, by means of a Płonka sum. This construction was first introduced and studied in [10,11,12]; for more recent expositions see [13] and [14]. Given a compatible family of homomorphisms between algebras of the same type {ϕ ij : A i → A j : i j}, indexed by the order of a join-semilattice (I, ∨), its Płonka sum is the algebra S defined on the disjoint union of their universes S = i∈I A i , so that for every nonconstant n-ary operation symbol σ and elements…”

Section: Proofmentioning

confidence: 99%

The Structure of Locally Integral Involutive Po-monoids and Semirings

Gil-Férez

Jipsen

Lodhia

2023

Lecture Notes in Computer Science

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We show that every locally integral involutive partially ordered semigroup (ipo-semigroup) A = (A, , •, ∼, −), and in particular every locally integral involutive semiring, decomposes in a unique way into a family {A p : p ∈ A + } of integral ipo-monoids, which we call its integral components. In the semiring case, the integral components are unital semirings. Moreover, we show that there is a family of monoid homomorphisms Φ = {ϕ pq : A p → A q : p q}, indexed over the positive cone (A + , ), so that the structure of A can be recovered as a glueing Φ A p of its integral components along Φ. Reciprocally, we give necessary and sufficient conditions so that the Płonka sum of any family of integral ipo-monoids {A p : p ∈ D}, indexed over a join-semilattice (D, ∨) along a family of monoid homomorphisms Φ is an ipo-semigroup.

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Section: Proofmentioning

confidence: 99%

The Structure of Locally Integral Involutive Po-monoids and Semirings

Gil-Férez

Jipsen

Lodhia

2023

Lecture Notes in Computer Science

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“…Considering classes defined by regular identities is part of research on identities, in which the structure of terms creating them matters. Example of such identities are: normal, externally compatible, P-compatible, biregular, symmetric and many others (see for example [5,11,12,14,15,16,18,19,22,24]). In recent years, Płonka sums have been analyzed not only within universal algebra but also logic.…”

Section: Introductionmentioning

confidence: 99%

On some extensions of the class of MV-algebras

Mruczek-Nasieniewska

2015

LLP

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In the present paper we will ask for the lattice L(MV Ex) of subvarieties of the variety defined by the set Ex(MV) of all externally compatible identities valid in the variety MV of all MV-algebras. In particular, we will find all subdirectly irreducible algebras from the classes in the lattice L(MV Ex) and give syntactical and semantical characterization of the class of algebras defined by P-compatible identities of MV-algebras.

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The lattice of subvarieties of the biregularization of the variety of Boolean algebras

Płonka

2001

Discuss. Math. - General Algebra and Applications

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Let τ : F → N be a type of algebras, where F is a set of fundamental operation symbols and N is the set of all positive integers. An identity ϕ ≈ ψ is called biregular if it has the same variables in each of it sides and it has the same fundamental operation symbols in each of it sides. For a variety V of type τ we denote by V b the biregularization of V , i.e. the variety of type τ defined by all biregular identities from Id (V ).Let B be the variety of Boolean algebras of type τ b : {+, ·, } → N , where τ b (+) = τ b (·) = 2 and τ b ( ) = 1. In this paper we characterize the lattice L(B b ) of all subvarieties of the biregularization of the variety B.

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Subdirectly irreducible generalized sums of upper semilattice ordered systems of algebras

Abstract: In [15] the generalized sum of an upper (F 1 , F 2 )-semilattice ordered system of algebras was defined. In this paper we find necessary and sufficient conditions under which this construction yields subdirectly irreducible algebras.

Cited by 3 publications

References 8 publications

The Structure of Locally Integral Involutive Po-monoids and Semirings

The Structure of Locally Integral Involutive Po-monoids and Semirings

On some extensions of the class of MV-algebras

The lattice of subvarieties of the biregularization of the variety of Boolean algebras

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