Residuated EQ-algebras may not be residuated lattices

摘要

This paper aims to solve the question "whether there is a residuated EQ-algebra, which is not a residuated lattice?" In this paper, we first introduce the notion of residuated meet semilattice monoids and research some properties of them. We discuss the relations between residuated meet semilattice monoids and residuated lattices, and prove that every complete (or prelinear) residuated meet semilattice monoid is a residuated lattice. Especially, we construct a residuated meet semilattice monoid, which has no lattice structure. Furthermore, we discuss the relations between residuated meet semilattice monoids and residuated EQ-algebras. Finally by use of residuated meet semilattice monoids, we give a theorem to show that there are some residuated EQ-algebras, but they are not residuated lattices. We also give an example, which satisfies the conditions of the above theorem, and hence it is a residuated EQ-algebra but it is not a residuated lattice. (C) 2019 Elsevier B.V. All rights reserved.