The lattice of varieties of some ternary modes.
by
Agata Pilitowska
Warsaw Technical University, Department of Mathematics, 00-661 Warsaw, Poland

Let R be a commutative ring with unity and let ESM be the set of submodules of the R-module (E, +, R).

For each element r of R, define a ternary operation r by

r:ESM×ESM×ESM --> ESM;          (U, V, W) --> r(U, V, W):=U+rV+rW,

and consider the algebra (ESM, R) with the set R={r:r in R} of operations.

Since among the operations R is a semilattice operation, the algebras (ESM, R) are semilattice modes (A mode is an idempotent algebra, in the sense that each singleton is a subalgebra, and an entropic algebra, i.e. each operation as a mapping from a direct power of the algebra into the algebra is a homomorphism.), and the variety V(MS(R)) they generate forms a semilattice mode variety.

General semilattice modes were investigated by K. Kearnes. He has shown that to each variety V of semilattice modes one can associate certain commutative semiring S(V), which plays a similar rôle as the ring of a variety of affine spaces. Similarly as in the case of affine R-spaces, the lattice of subvarieties of a variety of semilattice modes is determined by the congruences of the associated semiring.

The semiring associated with V(MS(R)) is isomorphic to the semiring of finitely generated ideals of the ring R.

The topic of the talk will describe the lattice of all subvarieties of the variety V(MS(Z_n)).

Date received: May 24, 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 # caee-71.