Free two-sided join-semirings
by
Marcel Erné
Universität Hannover

As a natural extension of ideals in (distributive) lattices, we introduce so-called v-ideals and v-radicals in algebras of arbitrary (not necessarily finitary) type v with at least one binary operation. The v-ideals prove to be just the kernels of homomorphisms into so-called two-sided v-semirings, a common generalization of two-sided join-semirings and prequantales (a modern abstraction of ideal semirings). Similarly, the v-radicals are precisely the kernels of homomorphisms into v-frames, generalizing both distributive lattices and frames (locales). The v-ideals (resp. v-radicals) of a given v-algebra G form the free two-sided prequantale (resp. locale) over G, and the principal v-ideals (v-radicals) form the universal two-sided v-semiring (v-frame) quotient of G. These general reflection theorems provide a whole bunch of convenient constructions of free distributive structures in quite diverse varieties of algebras.

Date received: May 22, 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-52.