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Subdirect decompositions of algebras from 2-clone extensions of varieties

J. Płonka

Colloquium Mathematicae (1998)

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

Abstract

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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 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)≠ ∅ . 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.

How to cite

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Płonka, 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 -

References

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  14. [14] J. Płonka, Biregular and uniform identities of bisemilattices, Demonstratio Math. 20 (1987), 95-107. Zbl0679.08005
  15. [15] J. Płonka, On varieties of algebras defined by identities of some special forms, Houston J. Math. 14 (1988), 253-263. Zbl0669.08006
  16. [16] J. Płonka, Biregular and uniform identities of algebras, Czechoslovak Math. J. 40 (1990), 367-387. Zbl0728.08005
  17. [17] J. Płonka, P-compatible identities and their applications to classical algebra, Math. Slovaca 40 (1990), 21-30. Zbl0728.08006
  18. [18] J. Płonka, Clone compatible identities and clone extensions of algebras, ibid. 47 (1997), 231-249. Zbl1018.08001
  19. [19] J. Płonka, Free algebras over n-clone extensions of n-downward regular varieties, in: General Algebra and Applications in Discrete Mathematics, Shaker Verlag, Aachen, 1997, 159-167. Zbl0964.08008
  20. [20] J. Płonka, On n-clone extensions of algebras, Algebra Universalis, in print. Zbl0935.08001

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