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The lattice of subvarieties of the biregularization of the variety of Boolean algebras

100%

Płonka J.

Discussiones Mathematicae - General Algebra and Applications

2001

tom 21

nr 2

255-268

Let τ: F → N be a type of algebras, where F is a set of fundamental operation symbols and N is the set of all positive integers. An identity φ ≈ ψ is called biregular if it has the same variables in each of it sides and it has the same fundamental operation symbols in each of it sides. For a variety V of type τ we denote by $V_{b}$ the biregularization of V, i.e. the variety of type τ defined by all biregular identities from Id(V). Let B be the variety of Boolean algebras of type $τ_{b}: {+,·,´} → N$, where $τ_{b}(+) = τ_{b}(·) = 2$ and $τ_{b}(´) = 1$. In this paper we characterize the lattice $ℒ(B_{b})$ of all subvarieties of the biregularization of the variety B.

Left-outtermost extension of some varieties

88%

Płonka J.

2003

tom Vol. 36, nr 1

37--51

Let r : F - > N be a type of algebras F is a nonempty set of fundamental operation symbols and N is the set of all positive integers. An identity ip fa if) of type T we call left-outermost if the left-outermost variables in ip and ip are the same. For a variety V of type r we denote by Vi the variety of type r defined by all left-outermost identities from Id(V). Vl is called the left-outermost extension of V. In this paper we study minimal generics, subdirectly irreducible algebras and lattices of subvarieties in left-outermost extensions of some generalizations of the variety D of all distributive lattices.

Subdirect decompositions of algebras from 2-clone extensions of varieties

75%

Płonka J.

Colloquium Mathematicum

1998

tom 77

nr 2

189-199

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.

Subdirect product representations of some unary extensions of semilattices

63%

Płonka J.

2008

tom Vol. 41, nr 2

263-272

An algebra [...] represents the sequence so = (0, 3, l, l, . . .) if there are no constants in [...], there are exactly 3 distinct essentially unary polynomials in [...] and exactly l essentially n-ary polynomial in [...] for every n > l . It was proved in [4] that an algebra [..] represents the sequence so if and only if it is clone equivalent to a generic of one of three varieties V1, V2, V3, see Section l of [4]. Moreover, some representations of algebras from these varieties by means of semilattice ordered systems of algebras were given in [4] . In this paper we give another, by subdirect products, representation of algebras from V1, V2, V3. Moreover, we describe all subdirectly irreducible algebras from these varieties and we show that if an algebra [...] represents the sequence so, then it must be of cardinality at least 4.