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

Colloquium Mathematicae (1998)

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

## Access Full Article

top## Abstract

top## How to cite

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 -

## References

top- [1] R. Balbes, A representation theorem for distributive quasilattices, Fund. Math. 68 (1970), 207-214. Zbl0198.33504
- [2] S. Burris and H. P. Sankappanavar, A Course in Universal Algebra, Springer, New York, 1981.
- [3] I. Chajda, Normally presented varieties, Algebra Universalis 34 (1995), 327-335.
- [4] E. Graczyńska, On normal and regular identities, ibid. 27 (1990), 387-397. Zbl0713.08007
- [5] G. Grätzer, Universal Algebra, 2nd ed., Springer, New York, 1979.
- [6] B. Jónsson and E. Nelson, Relatively free products in regular varieties, Algebra Universalis 4 (1974), 14-19. Zbl0319.08002
- [7] H. Lakser, R. Padmanabhan and C. R. Platt, Subdirect decomposition of Płonka sums, Duke Math. J. 39 (1972), 485-488. Zbl0248.08007
- [8] I. I. Mel'nik, Nilpotent shifts of varieties, Mat. Zametki 14 (1973), 703-712 (in Russian); English transl.: Math. Notes 14 (1973), 692-696.
- [9] J. Płonka, On distributive quasi-lattices, Fund. Math. 60 (1967), 191-200. Zbl0154.00709
- [10] J. Płonka, On a method of construction of abstract algebras, ibid. 61 (1967), 183-189. Zbl0168.26701
- [11] J. Płonka, On equational classes of abstract algebras defined by regular equations, ibid. 64 (1969), 241-247. Zbl0187.28702
- [12] J. Płonka, On sums of direct systems of Boolean algebras, Colloq. Math. 20 (1969), 209-214. Zbl0186.30802
- [13] J. Płonka, On the subdirect product of some equational classes of algebras, Math. Nachr. 63 (1974), 303-305. Zbl0289.08002
- [14] J. Płonka, Biregular and uniform identities of bisemilattices, Demonstratio Math. 20 (1987), 95-107. Zbl0679.08005
- [15] J. Płonka, On varieties of algebras defined by identities of some special forms, Houston J. Math. 14 (1988), 253-263. Zbl0669.08006
- [16] J. Płonka, Biregular and uniform identities of algebras, Czechoslovak Math. J. 40 (1990), 367-387. Zbl0728.08005
- [17] J. Płonka, P-compatible identities and their applications to classical algebra, Math. Slovaca 40 (1990), 21-30. Zbl0728.08006
- [18] J. Płonka, Clone compatible identities and clone extensions of algebras, ibid. 47 (1997), 231-249. Zbl1018.08001
- [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] J. Płonka, On n-clone extensions of algebras, Algebra Universalis, in print. Zbl0935.08001

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.