Commutative Idempotent Residuated Lattices

We investigate the variety of residuated lattices with a commutative and idempotent monoid reduct. A residuated lattice is an algebra A = (A,∨,∧, ·, e, /, \) such that (A,∨,∧) is a lattice, (A, ·, e) is a monoid and for every a, b, c ∈ A ab ≤ c ⇔ a ≤ c/b ⇔ b ≤ a\c. The last condition is equivalent to the fact that (A,∨,∧, ·, e) is a lattice-ordered monoid and for every a, b ∈ A there is a greatest c such that cb ≤ a (denoted a/b) and a greatest d such that bd ≤ a (denoted b\a). It is easy to see that the class RL of all residuated lattices is a variety. We are concerned about the variety CIdRL of commutative idempotent (CI) residuated lattices, i.e. the subvariety of RL given by equations xy ≈ yx and xx ≈ x. In other words, residuated lattices whose semigroup reduct is a semilattice. For example, every Heyting algebra is a CI residuated lattice, where ab = a ∧ b and a/b = b\a = b→ a for every a, b (see e.g. [3], p. 30). Foundation of the theory of residuated lattices goes as far as to 1930’s, when Dilworth and Ward [5] studied lattices of ring ideals. A recent introduction can be found in [4] and [10] and commutative residuated lattices were particularly studied in [9]. We will use the notation and terminology of these papers. We also assume a basic familiarity with universal algebra, standard references are [3] and [12]. In CI residuated lattices, we drop the operation \, since under commutativity x/y ≈ y\x. The lattice order will be denoted by ≤. We put a b iff ab = a; hence is the semilattice order, where · is regarded as the meet; e is its top element. When refering to an order, we mean the lattice order ≤, unless explicitly stated otherwise. We put A = {a ∈ A : a ≥ e} and A− = {a ∈ A : a ≤ e} and we call A the positive cone and A− the negative cone of A (regarded as lattice-ordered monoids; indeed, they may not be closed on residuation). The bottom element (in the lattice order) is denoted 0 and the top element is denoted 1, if they exist; it is easy to see that, in any residuated lattice, if 0 exists, then 1 exists, 0a = a0 = 0 and a/0 = 1/a = 1 (see also [4]); particularly, 0 is also the bottom element of the semilattice order in any CI residuated lattice. 1991 Mathematics Subject Classification. 06F05.