Abstract

In this paperAlizadeh, M., residuated expansionsResiduated expansion of lattice-ordered structures are explored, in particular, of both lattice-ordered groupoidslattice-ordered groupoid and lattices with implication. Here, a residuated expansionResiduated expansion is an expansion in which the law of residuation between fusion and implication holds. Thus, residuated expansionsResiduated expansion discussed here take the form of residuated lattice-ordered groupoidslattice-ordered groupoid. A necessary and sufficient condition is given for a lattice-ordered groupoid and also for a lattices with implication to be expandable to a residuated one. Then, our attention is focused to the case where these lattice-ordered structures are bounded and distributive. Each of these structures is shown to be embedded into a residuated one in most cases. Weak HeytingHeyting, A. algebras are algebras for subintuitionistic logicsSub-intuitionistic logics, which are special bounded distributive lattices withDistributive lattices with operators implication. By applying the above result to them, it is shown that every weak HeytingHeyting, A. algebra can be embedded into the canonical residuated expansionResiduated expansion. This establishes a close link betweenWeak Heyting algebra weak Heyting algebrasHeyting, A. and the residuated ones, which isWeak Heyting algebra examined in more detail for the finite embeddability property and the amalgamation property.