Algebras with hyperidentities of the variety of Boolean algebras

“…In order to explain the difference of the notions invented above with the notions of Yu. Movsisyan [32]- [35] we invent the following: Proposition 5.5. Let a type τ and the monoid H(τ ) = (H(τ ), •, σ id ) of all hypersubstitutions of type τ be given.…”

Section: M-hyperquasi-identitiesmentioning

confidence: 99%

M-Hyperquasivarieties

Graczyńska¹,

Schweigert²

2006

Demonstratio Mathematica

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Abstract.We consider the notion of M-hyper-quasi-identities and M-hyperquasivarieties, as a common generalization of the concept of quasi-identity (hyper-quasiidentity) and quasivariety (hyper-quasivariety) invented by A. I. Mal'cev, cf. [13], cf.[6] and hypervariety invented by the authors in [15], [8] and hyperquasivariety [9]. The results of this paper were presented on the 69th Workshop on General Algebra, held at Potsdam University (Germany) on March 18-20, 2005. NotationsAn identity is a pair of terms where the variables are bound by universal quantifiers. Let us take the following medial identity as an example VuVxVi/Vw (u • x) • (y • w) = (u • y) • (x • w).Let us look at the following hyperidentity VFVuVxVyVw F{F{u, v), F{x, y)) = F (F(u, x),F(v, y)).The hypervariable F is considered in a very specific way. Firstly every hypervariable is restricted to functions of a given arity. Secondly F is restricted to term functions of the given type. Let us take the variety A n $ of abelian groups of finite exponent n. Every binary term t = t(x,y) can be presented by t(x, y) = ax + by with a, b € No-If we substitute the binary hypervariable F in the above hyperidentity by ax + by, leaving its variables unchanged, we get a(au + bv) + b(ax + by) = a(au + bx) + b(av + by).This identity holds for every term t(x,y) = ax + by for the variety A n0 . Therefore we say that the hyperidentity holds for the variety A n fi. where U = Si are k-ary identities of a given type, for i = 0,..., n. A qausi-identity above is satisfied in an algebra A of a given type if and only if the following implication is satisfied in A: given a sequence ai,..., a^ of elements of A. If this elements satisfy the equations U(ai,... ,afc) = Si(ai,..., a n ) in A, for % -0,1,..., n-1, then the equality t n (ai,..., afc) = s n (ai,..., afc) is satisfied in A. In that case we write:A |= (t0 = so) A ... A (tn_i = sn_i) -> (tn = s n ).A quasi-identity e is satisfied in a class V of algebras of a given type, if and only if it is satisfied in all algebras A belonging to V. In that case we write:V |= (to = so) A ... A (tn-1 = S"_l) f (u,xi,...,xn) = f (u,yi,...,yn) f (v,xi,...,xn) = f(v,xi,...,xn) A variety V is called abelian, if each algebra of V is abelian. We can derive by the associative hyperidentity: F(x,y),z), that the following hyper-quasi-identity holds in RB : A lattice is semidistributive if it is simultaneously join and meet semidistributive. Semidistributive lattices PROPOSITION 2.3. Let L = (L, A, V) be a lattice which is semidistributive. Then the following hyper quasi-identity is hypersatisfied in L: (F(:r, y) = F(x, z)) -(F(x, y) = F(x, G(y, z)))Proof. We consider all cases on hypervariables of a semidistributive lattice. of type r such that B = A" 7 = (^4, to, ii,..., ...) with t 7 being equal to the term cr(/ 7 (xo,..., x T (/)_i)) for a fixed a E M.For a class K of algebras of type r we denote by DM(^0 the class of all M-derived algebras (of type r) of K. A class K is called M -deriverably closed if and only if DM (A") C ...

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“…Therefore up to the paper of Padmanabhan and Penner [22] it was not known what are hyperidentities of (distributive) lattices. Assuming a quite different definition of a hyperidentity Movsisyan [20] has been engaged in studying the same problems as Padmanabhan and Penner [22], but also for Boolean algebras (see [19]). Padmanabhan and Penner's not easy results stimulated the author for continuation this kind of examinations in some generalizations of (distributive) lattices.…”

Section: Introductionmentioning

confidence: 99%