Construction of finite L-groups

In this paper we give some necessary and sufficient conditions for a fuzzy relation to be a compatible fuzzy equality of some L-group in the sense of Bělohlávek. These characterizations enable practical applications for construction of L-groups with a given skeleton. The construction has two steps: determination of the general form of fuzzy pre-equalities on a group and then solving a system of inequalities in the given residuated lattice. Some concrete examples of construction of finite L-groups with the given skeleton are presented.