Lattice of fuzzy subalgebras in universal algebra. (Q1771915)

This paper is devoted to the so-called fuzzy subalgebras of universal algebras. A fuzzy subalgebra of an algebra \(A\) is a fuzzy subset of \(A\) which is closed (in a natural way generalizing the non-fuzzy case) with respect to the operations of \(A\). The authors show a bijective correspondence between fuzzy subalgebras of \(A\) and so-called weights over \(A\) (which are certain mappings from \(\text{Sub}(A)\) to \([ 0,1 ] \)). Then they describe fuzzy subalgebras generated by fuzzy subsets by means of weights. The main results of the paper consist in showing that (1) the complete lattice of all subalgebras of \(A\) and the complete lattice of fuzzy subalgebras of \(A\) satisfy the same identities; and (2) the same is true for quasi-identities. These results are then applied to particular classes of algebras and systems of their fuzzy subalgebras.