Generalized transformation semigroups having proper dense subsemigroups

Abstract

For any sets X, Y, let 𝒯 (X, Y) = the set of all mappings from X into Y, 𝒫𝒯 (X, Y) = the set of all mappings from subsets of X into Y and ℐ (X, Y) = the set of all 1-1 mappings from subsets of X in Y. If 𝒮 (X, Y) is any one of the sets 𝒯 (X, Y), 𝒫𝒯(X, Y) or ℐ (X, Y) and θ ɛ 𝒮 (X, Y) let (𝒮 (X, Y), θ) denote the semigroup (𝒮 (X, Y),*) with α*β = αθβ for all α, β ɛ 𝒮 (X, Y). The following theorem is the main result of this research. Theorem. Let X and Y be sets. Let 𝒮 (X, Y) denote any one of the sets 𝒯 (X, Y), 𝒫𝒯(X, Y) or ℐ (X, Y) and let θ ɛ 𝒮 (X, Y). Then (𝒮 (X, Y),θ) has a proper dense subsemigroup if and only if X and Y and both infinite and |𝛁θ| = min {|X|, |Y|} where 𝛁θ is the range of θ. If X and Y are both infinite and |𝛁θ| = min {|X|, |Y|}, then the following statements hold : (1) Suppose A is an infinite subset of 𝛁θ such that | 𝛁θ-A| = | 𝛁θ |. For each a ɛ A, choose yₐ ɛ aθ̄¹. Then the set U defined by U = {α ɛ 𝒮 (X, Y)| |Aα ⋂ (Y-{yₐ | a ɛ A}) | < |A|} (𝒮 (X, Y),θ). (2) (𝒮 (X, Y),θ) has infinitely many proper dense subsemigroups and the cardinality of the collection of such proper dense subsemigroups in not less than min {|X|, |Y|}.