Finite lattices as lattices of R-congruences of finite unars and Abelian groups
by
A. M. Nurakunov
Institute of Mathematics, National Academy of Sciences, Kyrgyz Republic

Let R be quasivariety of algebras. A congruence \theta on algebra A is called R-congruence if A\\theta belongs to R. A set Con_RA of all R-congruences of algebra A is algebraic lattice and is called a lattice of R-congruences on algebra A. If R is variety then this lattice is usual lattice of congruences ConA. It's well known problem [1]: Is any finite lattice is isomorphed to a lattice of congruence of finite algebra? The problem is still open. There exist a lot of results concerning this problem. We proof that the problem have positive solution for lattices of R-congruences.

THEOREM 1. Let L be finite lattice. Then there exist locally finite quasivariety R (K) of unars (groups) and finite unar U (group G) such that L is isomorphed to Con_RU (Con_KA)       .

Subclass K of quasivariety R of algebras is called R-variety if K=R \cap V for some variety V. A set L_V(R) of all R-varieties of quasivariety R is co-algebraic lattice and is called R-varieties lattice.If R is variety then this lattice is usual varieties lattice. In [2] was prooved that there exist locally finite quasivariety R of unar algebras with two unar operations such that  L is isomorphed to L_V(R).

THEOREM 2. Let L be finite lattice. Then there exist locally finite quasivariety R (K) of unars (groups) such that  L is isomorphed to L_V(R) (L_V(K)).

1. G.Gratzer, General Theory of Lattices , Verlag, Basel 1978.

2. K.V.Adaricheva, V.A.Gorbunov, Equational closure operator and forbbiden semidistributive lattices, Sibirskii matematicheskii jurnal, 30(6), 1989, pp.7-25.

Date received: April 9, 2002

Copyright © 2002 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 # caiv-11.