On externary compatible identities
by
Ewa Graczyñska
Institute of Mathematics, Technical University of Opole. Poland.

We consider well known operators N and R on varieties V of a given type by defining N(V)=Mod(N(V)), R(V)=Mod(R(V)), where N(V) denotes the set of all "normal" identities satisfied in V, R(V) denotes the set of all "regular" identities satisfied in V, cf. [5]. We deal with some other types of identities and covarieties as well.

E(V) denotes the set of all identities (of a given type) satisfied in V. We recall the notion of "regular" and "normal" variety in the following way:

Definition 1: A variety V is normal (regular) if and only if the two-element zero-semigroup (two element sup-algebra of a given type) belongs to V.

The definition is equivalnt to the well known one, invented by I. I. Melnik and J. Plonka, which say that: a variety V is normal (regular) iff E(V)=N(V), (E(V)=R(V) respectively).

Externally compatible identities in algebras were defined by J. Plonka, see W. Chromik in [4].

We recall the definition from K. Gajewska-Kurdziel and K. Mruczek, cf. [2]: Given type (2, 1) of abelian groups with two fundamental operations: the sum x+y and inverse -x.

Definition[K. Gajewska-Kurdziel, K. Mruczek] 2001/2002: The identity of type (2, 1) is externary compatible if it is one of the form: x=x, p+q=r+s, -p=-q, for some terms p, q, r, s of type (2, 1) and x is a variable.

Definition[W. Chromik, K, Halkowska] 1991: An identity p=q of type (2,2)is externary compatible if it is of the form x = x, for a variable x or p and q are terms different of a variable and both have the same outermost functional symbol.

For a given variety V, Ex(V) denotes the set of all externary compatible identities satisfied in V.

Remark 1. The trivial identity: x=y, for different variabels x and y is not externary compatible, but in groups it is equivalent to an externary compatible one, for example: (z-z) + x = (z-z)+ y, where z is a variable (or a term). Remark 2. Given any identity p = q of type (2, 1). Then in the theory G of groups of type (2, 1) it is equivalent to an externary compatible identity, for example: (z - z) + p = (z - z) + q, where z is a variable (or a term).

THEOREM. Given a variety V of lattices of type (2,2). Then the lattice L(Ex(V)) is isomorphic to the direct product of the lattice L(V) and a three-element chain.

References: [1] K. Gajewska-Kurdziel, K. Mruczek, Sets of identities satisfied in abelian groups, Demonstratio Mathematica, vol. 35, No. 3, 2002, 447-453.

[2] K. Halkowska, B. Cholewinska, R. Wiora, Externary compatible identities of Abelian groups, Acta. Univ. Wratislav. No 1890, 1997, 163-170.

[3] W. Chromik, K. Halkowska, Subvarieties of the variety defined by externary compatible identities of distributive lattices, Demonstratio Mathematica, vol. XXIV, No. 1-2, 1991, pp. 235-240.

[4] W. Chromik, Externary compatible identities of algebras, Demonstratio Mathematica 23, 1990, 344-355.

[5] Ewa Graczynska, Universal algebra via tree operads, Oficyna Wydawnicza Politechniki Opolskiej, 2000.

Date received: January 30, 2003