Mokrytskyi T. Semigroups of partial order isomorphisms of partially ordered spaces.

The thesis is devoted to the study the algebraic properties of the monoid IPF(N^n) of order isomorphisms between principal filters of the finite power of the set of positive integers with the product order, for any positive integer n ≥ 2 and the algebraic properties of the monoid IPF(^\kappa{N}) of order isomorphisms between principal filters of the set ^\kappa{N} with the product order, for any infinite cardinal \kappa. Also the thesis is devoted to the topologization of the semigroup IPF(N^n). It is shown that the semigroup IPF(N^n) is bisimple, E-unitary and F-inverse semigroup. Green's relation, the semilattice of the idempotents and the natural partial order on the monoid IPF(N^n) is described. It is proved that the group of units H(I) of the monoid IPF(N^n) is isomorphic to the permutation group S_n, and maximal subgroups of this monoid are described. Also, it is proved that the semigroup IPF(N^n) is isomorphic to the semidirect product of the direct n-th power of the bicyclic monoid by the group of permutation S_n. It is shown that every non-identity congruence \mathfrak{C} on the semigroup IPF(N^n) is a group congruence. The least group congruence \mathfrak{C}_{mg} is described and it is proved that the quotient-semigroup IPF(N^n)/\mathfrak{C}_{mg} by the least group congruence \mathfrak{C}_{mg} is isomorphic to the semidirect product S_n ⋉ Z^n_+ of the direct n-th power of the additive group of integers Z^n_+ by the group of permutations S_n. It is shown that each Hausdorff shift-continuous topology on the semigroup IPF(N^n) is discrete. It is proved that if for some positive integer n ≥ 2 the semigroup IPF(N^n) is a dense subset of the Hausdorff semitopological semigroup (S,*) and I=S\IPF(N^n)\neq\varnothing, then I is a two-sided ideal in S. Next, it is proved that if for some positive integer n ≥ 2, the Hausdorff topological semigroup S contains the semigroup IPF(N^n) as a dense subsemigroup, then the square S х S is not a feebly compact space. It is constructed an example of a non-discrete Hausdorff compact shift-continuous topology \tau_{Ac} on the semigroup IPF(N^n) with adjoined zero. It is justified why this topology \tau_{Ac} is the unique Hausdorff compact shift-continuous topology on IPF(N^n) with adjoined zero. And it is proved that the Hausdorff locally compact semitopological semigroup IPF(N^n) with adjoined zero is either compact or discrete. It is shown that the semigroup IPF(^\kappa{N}) is bisimple, E-unitary and F-inverse semigroup. It is described the semilattice of the idempotents, the natural partial order and Green's relations on the monoid IPF(^\kappa{N}). It is proved that the group of units H(I) of the monoid IPF(^\kappa{N}) is isomorphic to the group S_\kappa of all bijections of the cardinal \kappa, and maximal subgroups of this monoid are described, too. Also, it is proved that the semigroup IPF(^\kappa{N}) is isomorphic to the semidirect product S_\kappa ⋉ ^\kappa{B} of the semigroup ^\kappa{B} by the group S_\kappa. It is shown that every non-identity congruence \mathfrak{C} on the semigroup IPF(^\kappa{N}) is a group, described the least group congruence \mathfrak{C}_{mg} and proved that the quotient-semigroup IPF(^\kappa{N})/\mathfrak{C}_{mg} is isomorphic to the semidirect product S_\kappa ⋉ ^\kappa{Z}_+ of the group ^\kappa{Z}_+ by the group S_\kappa.