Countable Versus Uncountable Ranks in Infinite Semigroups of Transformations and Relations

Abstract

The relative rank rank(S : A) of a subset A of a semigroup S is the minimum cardinality of a set B such that A ∪ B = S. It follows from a result of Sierpiński that, if X is infinite, the relative rank of a subset of the full transformation semigroup T X is either uncountable or at most 2. A similar result holds for the semigroup B X of binary relations on X.A subset S of T N is dominated (by U ) if there exists a countable subset U of T N with the property that for each σ in S there exists µ in U such that iσ iµ for all i in N. It is shown that every dominated subset of T N is of uncountable relative rank. As a consequence, the monoid of all contractions in T N (mappings α with the property that |iα − jα| |i − j| for all i and j) is of uncountable relative rank.It is shown (among other results) that rank(B X : T X ) = 1 and that rank(B X : I X ) = 1 (where I X is the symmetric inverse semigroup on X). By contrast, if S X is the symmetric group, rank(B X : S X ) = 2.

Generating the Full Transformation Semigroup Using Order Preserving Mappings

“…Sierpiński [15] showed that any countable set of maps from X to X is contained in a 2-generated subsemigroup of T X . An alternative proof of this was given by Banach [1]; see also [8]. An immediate corollary of this result is that the relative rank of a subset of T X is either uncountable or at most two.…”

“…The corresponding result, that any countable set of permutations is contained in a 2-generated subgroup of S X , was given some years later in [3]. The analogues of these results in the semigroup of all binary relations, the semigroup of all partial maps, and the symmetric inverse semigroup were proven in [8].…”

“…In [8] the case where X = ‫ގ‬ (with the usual ordering) was considered, and it was shown that rank(T ‫ގ‬ : O ‫ގ‬ ) = 1.…”

Generating continuous mappings with Lipschitz mappings

“…For example, a classical result in the field, proved by Sierpiński, states that every countable set of mappings from an infinite set X to itself can be generated using two such mappings; see [1], [7] or [12]. The semigroup of all mappings from X to X is denoted by T X .…”

“…In [5], to mention another interesting result, it has been shown that whenever G ⊆ S X then rank(S X : G) is 0, 1 or uncountable. Slight modifications of Banach's proof of Sierpiński's result can be used to prove analogous results for binary relations, partial mappings, and partial bijections of infinite sets; see [7].…”