Submonoids of Groups, and Group-representability of Restricted Relation Algebras

Marek Kuczma asked in 1980 whether for every positive integer n, there exists a subsemigroup M of a group G, such that G is equal to the n-fold product M M−1M M−1 . . . M(−1) n−1 , but not to any proper initial subproduct of this product. We answer his question affirmatively, and prove a more general result on representing a certain sort of relation algebra by subsets of a group. We also sketch several variants of the latter result. M. Kuczma [4, Problem P190, p. 304] raised the question quoted in the Abstract for n = 3 in particular, and also for arbitrary n. The case n = 3 was answered affirmatively by an example of W. Benz [4, Remark P190S1, p. 305]. We sketch in §1 a construction that works for all n, then prove in §4 a general result, of which, as we note in §5.1, the behavior of that example is a consequence. In §6 we look at some variants (and possible further variants) of our general result. In particular, in §6.4 we note a class of operations on binary relations, described in B. Jónsson’s survey paper [2], which can be incorporated into that result. 1. Sketch of the construction answering Kuczma’s question If G is a group of permutations of a set X, we shall write elements of G to the right of elements of X, and compose them accordingly. Given a positive integer n, let X1, . . . , Xn be disjoint infinite sets, let X = X1 ∪ . . . ∪Xn, and let G be the group of those permutations g of X such that for all but finitely many x ∈ X, the element xg lies in the same set Xi as does x. Note that since the Xi are infinite, the (finite) number of elements carried into a given Xi by a given g ∈ G need not equal the (also finite) number moved out of it by g. Indeed, given any finite set of elements of X, and any assignment of a destination-set Xi for each of them, one can construct a g ∈ G which realizes these movements, and keeps all other elements in their original sets. (In achieving such a rearrangement, if the number of newcomers assigned to some Xi is not equal to the number of elements specified to leave Xi, then infinitely many elements have to be moved within Xi, as in “Hilbert’s Hotel” [1, p. 17], to accommodate these relocations.) Now let M ⊆ G be the submonoid consisting of those g which involve no movement of elements from one Xi to a different one, except for transitions from even-indexed sets X2i to adjacent odd-indexed sets, X2i−1 and X2i+1, as suggested by the picture (1) below. (For concreteness, n is there assumed even.) Dotted arrows show directed paths along which finitely many elements may move.