Axiomatisability Problems for S-systems

To a given type of algebraic systems # there corresponds at least one first order language L. One can then ask whether a property P, defined for members of #, is expressible in the language L. In other words, is there a set of sentences II such that a member M of # has property P if and only if all sentences in n are true in M. If the set II exists we say that P is definable in L. Further, Q) is axiomatisable in L and II axiomatises Si, where 2 is the subclass of # whose members have property P. In this paper we are concerned with the following problems. Given a monoid S, what conditions must S satisfy for the class of flat S-systems to be axiomatisable or for the class of projective S-systems to be axiomatisable? The corresponding questions for modules over a ring R have been fully answered by Eklof and Sabbagh in [4], The relevant algebraic definitions are given in Section 2, where we also outline some of the basic semigroup terms that we shall use. We do assume some knowledge of model theory, including the construction of ultraproducts. As far as possible we follow the notation and terminology of [7] for semigroup theory and [2] for model theory. We adopt the convention that an ordinal is the set of all smaller ordinals. If S is a monoid (R a ring with a 1) we let & denote the class of projective ^-systems (.R-modules) and SF the class of flat. S-sy stems CR-modules). A monoid or a ring is said to be perfect if & = &. In Theorem 3.1 we give necessary and sufficient conditions on a monoid S for the class 3F to be axiomatisable. Eklof and Sabbagh show that for a (unitary) ring R, & is axiomatisable if and only if $F is axiomatisable and R is perfect. To prove this they draw on a general result which is not true of 5-systems. We show that for a monoid S, if & is axiomatisable, then 2F is axiomatisable and 5 satisfies MR, the descending chain condition for principal right ideals. A ring satisfies MR if and only if it is perfect [1], but MR is not enough to give perfection for a monoid. It is shown in [5] that a monoid S is perfect if and only if it satisfies MR and condition A, which asserts that every 5-system satisfies the ascending chain condition for cyclic S'-subsystems. We prove in Proposition 4.3 that if & is axiomatisable, then a monoid S satisfies A if and only if it satisfies M, the ascending chain condition for principal left ideals. This enables us to deduce that in some fairly general cases, for example if S is regular, & is axiomatisable if and only if $F is axiomatisable and S is perfect. I should like to record my thanks to Dr J. B. Fountain for his generous advice with regard to this work.