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Subdirect decompositions of algebras from 2-clone extensions of varieties

100%

Płonka J.

Colloquium Mathematicum

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1998

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tom 77

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nr 2

189-199

EN

Let τ:F → ℕ be a type of algebras, where F is a set of fundamental operation symbols and ℕ is the set of nonnegative integers. We assume that |F|≥2 and 0 ∉ (F). For a term φ of type τ we denote by F(φ) the set of fundamental operation symbols from F occurring in φ. An identity φ ≉ ψ of type τ is called clone compatible if φ and ψ are the same variable or F(φ)=F(ψ)≠$\emptyset$. For a variety V of type τ we denote by $V^{c,2}$ the variety of type τ defined by all identities φ ≉ ψ from Id(V) which are either clone compatible or |F(φ)|, |F(ψ)|≥2. Under some assumption on terms (condition (0.iii)) we show that an algebra ${\gt A}$ belongs to $V^{c,2}$ iff it is isomorphic to a subdirect product of an algebra from V and of some other algebras of very simple structure. This result is applied to finding subdirectly irreducible algebras in $V^{c,2}$ where V is the variety of distributive lattices or the variety of Boolean algebras.

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Irredundant Decomposition of Algebras into One-Dimensional Factors

84%

Staruch B.

Bulletin of the Section of Logic

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2016

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tom 45

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nr 3/4

EN

We introduce a notion of dimension of an algebraic lattice and, treating such a lattice as the congruence lattice of an algebra, we introduce the dimension of an algebra, too. We define a star-product as a special kind of subdirect product. We obtain the star-decomposition of algebras into one-dimensional factors, which generalizes the known decomposition theorems e.g. for Abelian groups, linear spaces, Boolean algebras.

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Decomposition of Congruence Modular Algebras into Atomic, Atomless Locally Uniform and Anti-Uniform Parts

84%

Staruch B., Staruch B.

Bulletin of the Section of Logic

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2016

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tom 45

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nr 3/4

EN

We describe here a special subdirect decomposition of algebras with modular congruence lattice. Such a decomposition (called a star-decomposition) is based on the properties of the congruence lattices of algebras. We consider four properties of lattices: atomic, atomless, locally uniform and anti-uniform. In effect, we describe a star-decomposition of a given algebra with modular congruence lattice into two or three parts associated to these properties.

Ograniczanie wyników

2 Bulletin of the Section of Logic

1 Colloquium Mathematicum

2 Staruch B.

1 Płonka J.

2 2016

1 1998

Subdirect decompositions of algebras from 2-clone extensions of varieties

Irredundant Decomposition of Algebras into One-Dimensional Factors

Decomposition of Congruence Modular Algebras into Atomic, Atomless Locally Uniform and Anti-Uniform Parts