WEAK LATTICES

Abstract

The ordered set induced by a BCK-algebra \(\mathcal{A}\) can be equipped with a binary term operation on \(\mathcal{A}\) such that the resulting structure is a so-called weak semilattice.  If this structure is endowed with an antitone involution we can introduce a second binary operation and the structure arising this way is called a weak lattice.  Properties of weak lattices and weak semilattices are investigated and connections to directoids and semilattices are established.  Moreover, a derived structure similar to basic algebras is introduced and called a skew basic algebra.  An axiomatization of these algebras is presented.  It is shown that every bounded poset can be organized into a weak lattice and the number of non-isomorphic weak lattices of cardinality less than five is determined.