GMV Problem Description Tomasz Kowalski 2010-Feb-01 The general background is that of residuated lattices (RLs), namely, algebras that combine a lattice structure and a residuated monoid structure on the same universe. The two are linked by sharing the order: the natural lattice order coincides with the residuation order. For more on RLs, see [1], here we only recall a few things essential to the problem at hand. RLs are equationally defined by (x ^ y) ^ z = x ^ (y ^ z). (x V y) V z = x V (y V z). x ^ x = x. x V x = x. x ^ y = y ^ x. x V y = y V x. (x ^ y) V x = x. (x V y) ^ x = x. (x * ((x \ z) ^ y)) V z = z. ((y ^ (z / x)) * x) V z = z. (x \ ((x * y) V z)) ^ y = y. (((y * x) V z) / x) ^ y = y. (x * y) * z = x * (y * z). 1 * x = x. x * 1 = x. A particularly important subvariety of RLs is the variety of lattice-ordered groups (l-groups, LGs). These are defined, relative to RL, by (LG) x * (x \ 1) = 1. from which it follows that x \ 1 = 1 / x and so either of these can play the role of the group inverse. Another important subvariety is that of integral RLs (IRLs), defined by (I) x \ 1 = 1. which is equivalent to the requirement that the monoid unit be the top element. Three good examples of RLs come from the integers. Firstly, the integers under addition and the natural ordering form an l-group, where residuation is given by ordinary subtraction reversed, i.e., x \ y = y - x. Secondly, the non-positive integers under addition and truncated residuation form an integral RL. Thirdly, an interval [k,0] (k negative, of course) under truncated addition and truncated subtraction forms an integral RL which also happens to be an MV-algebra, a member of another important subvariety of RLs (MV stands for many-valued and goes back to \L{}ukasiewicz's many-valued logic). The three examples above have Generalised MV-algebras (GMVs) as a common generalisation. GMVs are defined by (GMV1) x V y = x / ((x V y) \ x). (GMV2) x V y = (x / (x V y)) \ x. In presence of integrality, GMV1 and GMV2 are respectively equivalent to (IGMV1) x V y = x / (y \ x). (IGMV2) x V y = (x / y) \ x. The last thing to observe before we can state the problem is that LGs and IGMVs are as far apart as they could be, namely, if an algebra is both an LG and an IGMV, then it is trivial. This suggests that GMVs should be somehow composed of an LG part and a IGMV part. Indeed, we have: THEOREM. GMV-algebras decompose as direct products of LGs and IGMVs. This is proved in [2] by semantic means. Our goal is to prove the same syntactically. To this end, we employ the observation that the varieties LG and IGMV are independent in the sense of [3], namely, there is a binary term x @ y, such that LG |= x @ y = x IGMV |= x @ y = y Indeed, it is easy to see that the term (x * (x \ 1)) * ((y \ 1) \ 1) satisfies the conditions. We can then axiomatise the varietal join of LG and IGMV by the following 15 equations: (1) x @ x = x. (2) (x @ y) @ z = x @ z. (3) x @ (y @ z) = x @ z. stating that the relations defined by (a,b) \in P1 iff a @ b = b and (a,b) \in P2 iff a @ b = a are reflexive and transitive (symmetry also follows) (4) (x ^ y) @ (z ^ u) = (x @ z) ^ (y @ u). (5) (x V y) @ (z V u) = (x @ z) V (y @ u). (6) (x \ y) @ (z \ u) = (x @ z) \ (y @ u). (7) (x / y) @ (z / u) = (x @ z) / (y @ u). stating that P1 and P2 preserve the operations, so they are congruences, and, finally (8) (x * (x \ 1)) @ 1 = x * (x \ 1). (9) 1 @ (x * (x \ 1)) = 1. (10) (x \ 1) @ 1 = 1. (11) 1 @ (x \ 1) = x \ 1. (12) (x / (y \ x)) @ (x V y) = x V y. (13) ((x / y) \ x) @ (x V y) = x V y. (14) (x V y) @ (x / (y \ x)) = x / (y \ x). (15) (x V y) @ ((x / y) \ x) = (x / y) \ x. stating that the P1-image is an LG, and the P2-image an IGMV. Notice how the LG axiom is "embedded" in (8) and (9), the I axiom in (10) and (11), and the IGMV axioms in (12)-(15). An equivalent, but slightly different, set of axioms is given in [4]. Now, to get the desired result, it suffices to prove that (1)-(15) follow from the GMV axioms. References [1] Galatos N., Jipsen P., Kowalski T., Ono H.: Residuated Lattices: An Algebraic Glimpse on Substructural Logics. Studies in Logic and the Foundations of Mathematics, vol. 151. Elsevier, Amsterdam (2007) [2] Galatos N., Tsinakis, C.: Generalized MV-algebras. Journal of Algebra \textbf{283} 254--291 (2005) [3] Gr\"{a}tzer G., Lakser H., P\l onka J.: Joins and direct products of equational classes. Canadian Mathematical Bulletin \textbf{12}, 741--744 (1969) [4] J\'{o}nsson B., Tsinakis C.: Products of classes of residuated structures. Studia Logica \textbf{77}, 267--292 (2004)