E-fuzzy groups

An E-fuzzy group is a lattice-valued algebraic structure, defined on a crisp algebra which is not necessarily a group. The crisp equality is replaced by a particular fuzzy one – denoted by E. The classical group-like properties are formulated as appropriate fuzzy identities – special lattice-theoretic formulas. We prove basic features of E-fuzzy groups: properties of the unit and inverses, cancellability, solvability of equations, subgroup properties and others. We also prove that for every cut of an E-fuzzy group, which is a classical subalgebra of the underlying algebra, the quotient structure over the corresponding cut of the fuzzy equality is a classical group.