On the disjunctive set problem

Abstract

A monoid M is called syntactic if it has a disjunctive subset, the latter being defined as a part P ⊆ M which is not a union of classes of a non-equality congruence on M. The SYNTACTIC MONOID problem asks to decide, for an arbitrary finite monoid M, whether or not M is syntactic. Our contribution to this problem is twofold: (1) We give an algorithm solving SYNTACTIC MONOID for a large class of finite monoids in O(¦M¦ 3) time (in O(¦M¦ 2) /oD = /oH ). (2) We show that a slight generalization of SYNTACTIC MONOID is NP-complete. This leaves us with the SYNTACTIC MONOID problem still open, but with its ‘hard core’ better circumscribed.