# Subdirect decompositions of algebras from 2-clone extensions of varieties

Colloquium Mathematicae (1998)

- Volume: 77, Issue: 2, page 189-199
- ISSN: 0010-1354

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topPłonka, J.. "Subdirect decompositions of algebras from 2-clone extensions of varieties." Colloquium Mathematicae 77.2 (1998): 189-199. <http://eudml.org/doc/210583>.

@article{Płonka1998,

abstract = {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 $\{A\}$ 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.},

author = {Płonka, J.},

journal = {Colloquium Mathematicae},

keywords = {lattice; varieties; subdirectly irreducible algebra; Boolean algebra; clone extension of a variety; subdirect product; subdirectly irreducible algebras; variety of distributive lattices; variety of Boolean algebras},

language = {eng},

number = {2},

pages = {189-199},

title = {Subdirect decompositions of algebras from 2-clone extensions of varieties},

url = {http://eudml.org/doc/210583},

volume = {77},

year = {1998},

}

TY - JOUR

AU - Płonka, J.

TI - Subdirect decompositions of algebras from 2-clone extensions of varieties

JO - Colloquium Mathematicae

PY - 1998

VL - 77

IS - 2

SP - 189

EP - 199

AB - 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 ${A}$ 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.

LA - eng

KW - lattice; varieties; subdirectly irreducible algebra; Boolean algebra; clone extension of a variety; subdirect product; subdirectly irreducible algebras; variety of distributive lattices; variety of Boolean algebras

UR - http://eudml.org/doc/210583

ER -

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