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## Colloquium Mathematicum

1998 | 77 | 2 | 189-199
Tytuł artykułu

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

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
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.
Słowa kluczowe
EN
Czasopismo
Rocznik
Tom
Numer
Strony
189-199
Opis fizyczny
Daty
wydano
1998
otrzymano
1997-05-28
Twórcy
autor
• Mathematical Institute of the Polish Academy of Sciences, Kopernika 18, 51-617 Wrocław, Poland
Bibliografia
• [1] R. Balbes, A representation theorem for distributive quasilattices, Fund. Math. 68 (1970), 207-214.
• [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.
• [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.
• [7] H. Lakser, R. Padmanabhan and C. R. Platt, Subdirect decomposition of Płonka sums, Duke Math. J. 39 (1972), 485-488.
• [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.
• [10] J. Płonka, On a method of construction of abstract algebras, ibid. 61 (1967), 183-189.
• [11] J. Płonka, On equational classes of abstract algebras defined by regular equations, ibid. 64 (1969), 241-247.
• [12] J. Płonka, On sums of direct systems of Boolean algebras, Colloq. Math. 20 (1969), 209-214.
• [13] J. Płonka, On the subdirect product of some equational classes of algebras, Math. Nachr. 63 (1974), 303-305.
• [14] J. Płonka, Biregular and uniform identities of bisemilattices, Demonstratio Math. 20 (1987), 95-107.
• [15] J. Płonka, On varieties of algebras defined by identities of some special forms, Houston J. Math. 14 (1988), 253-263.
• [16] J. Płonka, Biregular and uniform identities of algebras, Czechoslovak Math. J. 40 (1990), 367-387.
• [17] J. Płonka, P-compatible identities and their applications to classical algebra, Math. Slovaca 40 (1990), 21-30.
• [18] J. Płonka, Clone compatible identities and clone extensions of algebras, ibid. 47 (1997), 231-249.
• [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.
• [20] J. Płonka, On n-clone extensions of algebras, Algebra Universalis, in print.
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Bibliografia
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