Abstract:We give a description of the structure of the semigroups for which each principal ideal is a retract. The globally idempotent case is solved quickly using a-suitably modified-construction which has been developed by TuUy for the study of semigroups in which each ideal is a retract. The general case can be treated by a naturally obtained semigroup of subsets of the semigroup constructed for the globally idempotent case.
“…On the other hand, Tully [26] was the first to introduce a similar construction as above (with a special semilattice X and each I a being 0-simple) when he studied semigroups for which each ideal is a retract. Generalizing the results of Tully, the author [1] has shown that the class of all globally idempotent semigroups all of whose principal ideals being a retract is characterized by the construction S = (X;I a ,f a p ), X being a semilattice and each I a being 0-simple. The above mentioned construction of strict regular semigroups is useful for several purposes.…”
Section: A Semigroup S Is a Strict Regular Semigroup If And Only Ifmentioning
confidence: 84%
“…Next we give an interpretation of the ^-classes of F{3S v s/ ) r DEFINITION. Let / be a non-empty set and 5 be a semigroup with a unary operation xt-^x' 1 The main result of the present section is the following. THEOREM 3.9.…”
Section: Let S Be the Unary Subsemigroup Of F08j X Fs^l Which Is Genementioning
“…On the other hand, Tully [26] was the first to introduce a similar construction as above (with a special semilattice X and each I a being 0-simple) when he studied semigroups for which each ideal is a retract. Generalizing the results of Tully, the author [1] has shown that the class of all globally idempotent semigroups all of whose principal ideals being a retract is characterized by the construction S = (X;I a ,f a p ), X being a semilattice and each I a being 0-simple. The above mentioned construction of strict regular semigroups is useful for several purposes.…”
Section: A Semigroup S Is a Strict Regular Semigroup If And Only Ifmentioning
confidence: 84%
“…Next we give an interpretation of the ^-classes of F{3S v s/ ) r DEFINITION. Let / be a non-empty set and 5 be a semigroup with a unary operation xt-^x' 1 The main result of the present section is the following. THEOREM 3.9.…”
Section: Let S Be the Unary Subsemigroup Of F08j X Fs^l Which Is Genementioning
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