Directly indecomposables in semidegenerate varieties of connected po-groupoids (Q1013996)

A po-groupoid is a groupoid \((A,\cdot)\) such that the relation defined by \(x\leq y\) iff \(x\cdot y=x\) is a partial order on \(A\). A po-groupoid is said to be connected if the related partial order is connected. A variety \(V\) is called semidegenerate if no nontrivial member of \(V\) has a trivial subalgebra. The main result of the paper is Theorem 2. It states that if \(V\) is a semidegenerate variety of connected po-groupoids over a finite language, then the class of directly indecomposable algebras of \(V\) is axiomatizable by a \(\forall\exists\forall\exists\forall\exists\)-first-order sentence plus axioms for \(V\).