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

Colloquium Mathematicae (1998)

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

top

## Abstract

top
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)≠$\varnothing$. 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

top

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

top
1. [1] R. Balbes, A representation theorem for distributive quasilattices, Fund. Math. 68 (1970), 207-214. Zbl0198.33504
2. [2] S. Burris and H. P. Sankappanavar, A Course in Universal Algebra, Springer, New York, 1981.
3. [3] I. Chajda, Normally presented varieties, Algebra Universalis 34 (1995), 327-335.
4. [4] E. Graczyńska, On normal and regular identities, ibid. 27 (1990), 387-397. Zbl0713.08007
5. [5] G. Grätzer, Universal Algebra, 2nd ed., Springer, New York, 1979.
6. [6] B. Jónsson and E. Nelson, Relatively free products in regular varieties, Algebra Universalis 4 (1974), 14-19. Zbl0319.08002
7. [7] H. Lakser, R. Padmanabhan and C. R. Platt, Subdirect decomposition of Płonka sums, Duke Math. J. 39 (1972), 485-488. Zbl0248.08007
8. [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. [9] J. Płonka, On distributive quasi-lattices, Fund. Math. 60 (1967), 191-200. Zbl0154.00709
10. [10] J. Płonka, On a method of construction of abstract algebras, ibid. 61 (1967), 183-189. Zbl0168.26701
11. [11] J. Płonka, On equational classes of abstract algebras defined by regular equations, ibid. 64 (1969), 241-247. Zbl0187.28702
12. [12] J. Płonka, On sums of direct systems of Boolean algebras, Colloq. Math. 20 (1969), 209-214. Zbl0186.30802
13. [13] J. Płonka, On the subdirect product of some equational classes of algebras, Math. Nachr. 63 (1974), 303-305. Zbl0289.08002
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

top

## NotesEmbed?

top

You must be logged in to post comments.

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

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.