M-Hyperquasivarieties

Abstract

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 ...

Binary relations on the monoid of V-proper hypersubstitutions

“…In [8] the author determined the dimension of every subvariety of the variety RegB. This means that id p (V ) for these varieties is known.…”

“…Now we determine id p (V ) for every variety of bands. Since our proofs for subvarieties of RegB are quite different from the proofs in [8] we will give here the full proof. In [6] Proposition 4.1 was proved that for each variety of bands…”

Hyperidentities and Related Concepts, II

“…Starting with the 1960's the following second order formulae were studied in various domains of algebra and its applications (see [140], [141], [254], [255], [249], [250], [18], [64], [165], [168], [199], [21], [45], [110], [285], [24], [25], [218], [266], [170], [173], [178], [180], [272], [219], [267], [175], [195], [196], [197], [203], [211], [212], [120], [256], [144], [145], [222], [50], [51], [52], [53], [54], [63], [85], [86], [214], [215],…”