Separator Algebras
by
Arthur Knoebel

In unitary rings, one can characterize direct products internally in five different ways: central idempotents, factor ideals, orthogonal projections, factor congruences and decomposing functions. The two objectives of this talk are: (1) to show how to define direct products internally in these five ways for a much larger class of algebras; and (2) to prove embeddability of the new in the old. A separator algebra is an algebra having a quaternary term function q and two constants 0 and 1 satisfying q(1,0,x,y)=x and q(0,1,x,y)=y. Examples are unitary rings and bounded lattices: q(a,b,x,y)=ax+by. Any discriminator algebra is polynomially equivalent to a separator algebra, but not conversely. Theorem: Each separator algebra A has a Boolean algebra of factor elements Elem'A that is isomorphic to the Boolean algebra of factor congruences, and so isomorphic to the Boolean algebras of all the other factor objects. Theorem: For any separator algebra A there is a ring B, perhaps with additional operations, such that A is a subalgebra of a reduct of B; Elem'A is a subalgebra of Elem'B; and q(e,e',x,y)=ex+e'y for all e in Elem'B and all x,y in B. In a similar fashion, any separator algebra is embeddable in a distributive lattice or a subdirect product of discriminator algebras.

Date received: January 1, 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 # caig-43.