THE MONOID OF ORDER ISOMORPHISMS BETWEEN PRINCIPAL FILTERS OF sigma Nkappa
Taras Mokrytskyi · Visnyk Lvivskogo Universytetu Seriya Mekhaniko-Matematychna · 2022
Consider the following generalization of the bicyclic monoid. Let κ be any infinite cardinal and let ℐ𝒫ℱ(σℕ κ ) be the semigroup of all order isomorphisms between principal filters of the set σℕ κ with the product order. We shall study algebraic properties of the semigroup ℐ𝒫ℱ(σℕ κ ), show that it is bisimple, E-unitary, F-inverse semigroup, describe Green's relations on ℐ𝒫ℱ(σℕ κ ), describe the group of units H(𝕀) of the semigroup ℐ𝒫ℱ(σℕ κ ) and describe its maximal subgroups. We prove that the semigroup ℐ𝒫ℱ(σℕ κ ) is isomorphic to the semidirect product 𝒮 κ ⋉σℬ κ of the semigroup σℬ κ by the group 𝒮 κ , show that every non-identity congruence ℭ on the semigroup ℐ𝒫ℱ(σℕ κ )is a group congruence and describe the least group congruence on ℐ𝒫ℱ(σℕ κ ).