Abbott Groupoids

摘要

A concept of a groupoid satisfying certain properties common with Abbott implication algebras is introduced. It is shown that such a groupoid is a join semilattice with the greatest element 1 with respect to the induced ordering and and for each element p the interval [p, 1] is a lattice with an involutory antiautomorphism. Hence, it generalizes the concept of orthoimplication algebra, introduced by J.C.Abbott, and that of orthomodular implication algebra. We show that conversely every Abbott groupoid also can be introduced by such a semilattice.