Subdirectly Irreducible Squags of Cardinality \(3n\)

Abstract

In this paper, we construct a squag \(SQG(3n)\) of cardinality \(3n\) that contains three given arbitrary squags \(SQG(n)\)s as disjoint subquags. Accordingly, we can construct a subdirectly irreducible squag \(SQG(3n)\), for each \(n \geq 7\), with \(n \equiv 0, 3 \pmod{6}\). Also, we want to review the shape of the congruence lattice of non-simple squags \(SQG(n)\) for some \(n\) and to give a classification of the class of all \(SQG(21)\)s and the class of all \(SQG(27)\)s according to the shape of its congruence lattice. \(SQG(21)\)s are classified into three classes and \(SQG(27)\)s are classified into four classes. The construction of \(SQG(3n)\), which is given in this paper, helps us to construct examples of each class of both \(SQG(21)\)s and \(SQG(27)\)s.