On the Lattice of Collective Varieties of Commutative Exclusive Semigroups
by
Vladimir Dryomov
Volgodonsk, Russia

Let F_X be a free semigroup over an alphabet X. Let u be an element of F_X and V be a nonempty subset of F_X. A construction u in V will be called inclusive identity. We shall say that u in V is an inclusive identity of semigroup S if and only if for an arbitrary homomorphism j\colon F_X --> S, j(u) in j(V). Let \Phi be a system of identical inclusions. We shall call the class of all semigroups which satisfies \Phi a collective (or identical inclusive) variety [1].

The semigroup with the identical inclusion xyz in {xy, yz, xz} is called exclusive. It is known that the lattice of collective varieties of commutative exclusive semigroups has the cardinality of continuum [3]. The latice of all collective varieties of exclusive semilatices was described in [2]. In the present work all collective varieties of commutative semigroups with identical inclusion xyz in {xy, yz} are described.

Theorem. Any collective variety of commutative exclusive semigroups is one of the following:

\Gamma0=\Pi(xy=yx, xyz in {xy, yz}) \Gamma1=\Pi(xy=yx, xyz in {xy, yz}, xy in {xx, yy, xyz} ) \Gamma2=\Pi(xy=yx, xyz in {xy, yz}, xx in {x, xyz} ) \Gamma3=\Pi(xy=yx, xyz in {xy, yz}, xy in {xx, yy}) \Gamma4=\Pi(xy=yx, xyz in {xy, yz}, xy in {x, yy, xyz} ) \Gamma5=\Pi(xy=yx, xyz in {xy, yz}, x=xx ) \Gamma6=\Pi(xy=yx, xyz in {xy, yz}, xy in {xx, yy}, xy in {x, yy, xyz}) \Gamma7=\Pi(xy=yx, xyz in {xy, yz}, xy in {xx, xyz} ) \Gamma8=\Pi(xy=yx, xyz in {xy, yz}, xy in {xx, yy, xyz}, xx in {x, xyz}) \Gamma9=\Pi(xy=yx, xyz in {xy, yz}, xy in {x, yy} ) \Gamma10=\Pi(xy=yx, xyz in {xy, yz}, xy in {xx, yy}, xy in {xx, xyz} ) \Gamma11=\Pi(xy=yx, xyz in {xy, yz}, xy in {xx, yy}, xx in {x, xyz} ) \Gamma12=\Pi(xy=yx, xyz in {xy, yz}, xy in {xx, xyz}, xx in {x, xyz} ) \Gamma13=\Pi(xy=yx, xyz in {xy, yz}, x=xx, xy in {xx, yy, xyz}) \Gamma14=\Pi(xy=yx, xyz in {xy, yz}, xy in {xx, xyz}, xy in {x, yy} ) \Gamma15=\Pi(xy=yx, xyz in {xy, yz}, xy in {x, yy}, xx in {x, xyz} ) \Gamma16=\Pi(xy=yx, xyz in {xy, yz}, xy in {xx, yy}, xy in {xx, xyz}, xx in {x, xyz} ) \Gamma17=\Pi(xy=yx, xyz in {xy, yz}, xy in {xx, xyz}, x=xx ) \Gamma18=\Pi(xy=yx, xyz in {xy, yz}, xy in {xx, xyz}, xy in {x, yy}, xx in {x, xyz} ) \Gamma19=\Pi(xy=yx, xyz in {xy, yz}, xy in {x, yy}, x=xx ) \Gamma20=\Pi(xy=zt) \Gamma21=\Pi(xy=yx, xyz in {xy, yz}, xy in {xx, yy}, xy in {xx, xyz}, x=xx ) \Gamma22=\Pi(x=y)

[1] E. S. Lyapin. Generation of Semigroup Classes with the Aid of Homomorphisms. Semigroups and their homomorphisms, Leningrad, 1991, 1991, pp. 39-53 (in Russian)

[2] S. N. Bratchikov. Identical Inclusive Varieties of Semilatices. Modern Algebra, Vol. 2(22), Rostov-na-Donu, 1997, pp. 18-24 (in Russian)

[3] V. Dryomov. Example of Finite Semigroup without finite basis of collective identities. Modern Algebra. (to appear)

Date received: April 11, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caec-09.