The momoid of order isomorphisms between principal filters of σℕκ

Taras Mokrytskyi

Анотація


Consider the following generalization of the bicyclic monoid. Let κ be any infinite cardinal and let ℐ𝒫ℱ(σℕκ) be the semigroup of all order isomorphisms between principal filters of the set σℕκ with the product order. We shall study algebraic properties of the semigroup ℐ𝒫ℱ(σℕκ), show that it is bisimple, E-unitary, F-inverse semigroup, describe Green's relations on ℐ𝒫ℱ(σℕκ), describe the group of units H(𝕀)  of the semigroup ℐ𝒫ℱ(σℕκ) and describe its maximal subgroups. We prove that the semigroup ℐ𝒫ℱ(σℕκ) is isomorphic to the semidirect product 𝒮κ⋉σℬκ  of the semigroup  σℬκ  by the group 𝒮κ, show that every non-identity congruence ℭ on the semigroup ℐ𝒫ℱ(σℕκ)is a group congruence and describe the least group congruence on ℐ𝒫ℱ(σℕκ).

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