Journal of Algebra 247, 672–691 (2002) doi:10.1006/jabr.2001.9039, available online at http://www.idealibrary.com on
Representable Biresiduated Lattices C. J. van Alten1 Department of Mathematics, University of the Witwatersrand, Private Bag 3, Wits, 2050, South Africa E-mail:
[email protected] Communicated by Leonard Lipshitz Received February 27, 2001
INTRODUCTION A biresiduated lattice is an algebra A = A · / \ ∧ ∨ 1 such that A · 1 is a monoid, A ∧ ∨ is a lattice in which 1 is the maximum element, and / and \ are binary operations which satisfy the following “residuation” properties with respect to the lattice order: (∀ a b c ∈ A) a·c ≤b
iff c ≤ a\b
c·a≤b
iff
c ≤ b/a
These algebras were first studied by W. Krull in [Kru28] as abstractions of ideal lattices of rings enriched with the monoid operation of idealmultiplication. A recent investigation into the class of biresiduated lattices, which we denote by , has been undertaken by the author and W. J. Blok in [BV]. It is shown there, in particular, that arises naturally as a class of models of the full Lambek calculus extended by the rule of weakening; the operations / and \ correspond to right and left implication connectives, respectively. The objective of this paper is to axiomatize the class of all biresiduated lattices that may be represented as subalgebras of products of linearly ordered biresiduated lattices. Such biresiduated lattices are called representable and the class of all such algebras is denoted by . We shall prove 1 Funding for this research was provided by the National Research Foundation of South Africa.
672 0021-8693/02 $35.00 2002 Elsevier Science
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representable biresiduated lattices
673
that is axiomatized, relative to , by the identity x\y ∨ w · z\ y\x · z
/w = 1
(1)
or, equivalently, by x\y ∨ w/ w/ y\x \z \z
= 1 The second identity uses only the operations /, \, ∨, and 1, so we shall also obtain axiomatizations of the class of representable algebras in subreduct classes of whose languages contain /, \, ∨, and 1. Axiomatizations of the class of representable algebras in some related classes of algebras have been considered in the literature, notably the varieties of commutative biresiduated lattices, called residuated lattices, and lattice-ordered groups. The class of representable residuated lattices is axiomatized, relative to the variety of all residuated lattices, by the identity x\y ∨ y\x = 1. This result is contained in [Fle87], although an earlier result by M. Pałasinski in [Pał80] characterizes the representable BCKalgebras, which are the \ 1-subreducts of residuated lattices (see also [Raf87]). We obtain the above-mentioned result as a corollary to our main result and we show that x\y ∨ y\x = 1 alone does not characterize the representable biresiduated lattices. The negative cone of a lattice-ordered group is an example of a biresiduated lattice if we define y/x and x\y as y · x−1 ∧ 1 and x−1 · y ∧ 1, respectively, and the monoid operation as the group operation restricted to the negative cone. Such a biresiduated lattice is cancellative and complemented (see Section 5). Moreover, every cancellative and complemented biresiduated lattice arises from a lattice-ordered group in this way. It was shown by Lorentzen in [Lor49] (see also [AF88], for example) that the class of representable lattice-ordered groups is axiomatized, relative to the variety of all lattice-ordered groups, by the condition x
