A Residuated Lattice of L-Fuzzy Subalgebras of a Mono-Unary Algebra

Abstract

Given a complete residuated lattice [Formula: see text] and a mono-unary algebra [Formula: see text], it is well known that [Formula: see text] and the residuated lattice [Formula: see text] of [Formula: see text]-fuzzy subsets of [Formula: see text] satisfy the same residuated lattice identities. In this paper, we show that [Formula: see text] and the residuated lattice [Formula: see text] of [Formula: see text]-fuzzy subalgebras of [Formula: see text] satisfy the same residuated lattice identities if and only if the Heyting algebra [Formula: see text] of subuniverses of [Formula: see text] is a Boolean algebra. We also show that [Formula: see text] is a Boolean algebra (respectively, an [Formula: see text]-algebra) if and only if [Formula: see text] is a Boolean algebra (respectively, an [Formula: see text]-algebra) and [Formula: see text] is a Boolean algebra.