On transformation semigroups which are ℬ 𝒬 -semigroups

S. Nenthein and Y. Kemprasit

International Journal of Mathematics and Mathematical Sciences, 2006, vol. 2006, 1-10

Abstract:

A semigroup whose bi-ideals and quasi-ideals coincide is called a ℬ 𝒬 - semigroup . The full transformation semigroup on a set X and the semigroup of all linear transformations of a vectorspace V over a field F into itself are denoted, respectively, by T ( X ) and L F ( V ) . It is known that every regular semigroup is a ℬ 𝒬 -semigroup. Then both T ( X ) and L F ( V ) are ℬ 𝒬 -semigroups.In 1966, Magill introduced and studied the subsemigroup T ¯ ( X , Y ) of T ( X ) , where ∅ ≠ Y ⊆ X and T ¯ ( X , Y ) = { α ∈ T ( X , Y ) | Y α ⊆ Y } . If W is a subspace of V , the subsemigroup L ¯ F ( V , W ) of L F ( V ) will be defined analogously. In this paper, it is shown that T ¯ ( X , Y ) is a ℬ 𝒬 -semigroup if and only if Y = X , | Y | = 1 , or | X | ≤ 3 , and L ¯ F ( V , W ) is a ℬ 𝒬 -semigroup if and only if (i) W = V , (ii) W = { 0 } , or (iii) F = ℤ 2 , dim F V = 2 , and dim F W = 1 .

Date: 2006
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:012757

DOI: 10.1155/IJMMS/2006/12757

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