Algebra Commons^™

Commutative Doubly-Idempotent Semirings Determined By Chains And By Preorder Forests, Natanael Alpay, Peter Jipsen

Mathematics, Physics, and Computer Science Faculty Books and Book Chapters

A commutative doubly-idempotent semiring (cdi-semiring) (S,V,·,0,1) is a semilattice (S,V,0) with x V 0 = x and a semilattices (S,·,1) with identity 1 such that x0 = 0, and x(y V z) = xy V xz holds for all x, y, z ϵ S. Bounded distributive lattices are cdi-semirings that satisfy xy = x ^ y, and the variety of cdi-semirings covers the variety of bounded distributive lattices. Chajda and Länger showed in 2017 that the variety of all cdi-semirings is generated by a 3-element cdi-semiring. We show that there are seven cdi-semirings with a V-semilattice of height …

Weakening Relation Algebras And Fl2-Algebras, Nikolaos Galatos, Peter Jipsen

Mathematics, Physics, and Computer Science Faculty Books and Book Chapters

FL²-algebras are lattice-ordered algebras with two sets of residuated operators. The classes RA of relation algebras and GBI of generalized bunched implication algebras are subvarieties of FL²-algebras. We prove that the congruences of FL²-algebras are determined by the congruence class of the respective identity elements, and we characterize the subsets that correspond to this congruence class. For involutive GBI-algebras the characterization simplifies to a form similar to relation algebras.

For a positive idempotent element p in a relation algebra A, the double division conucleus image p/A/p is an (abstract) weakening relation algebra, …