The monoid of monotone injective partial selfmaps of the poset (N3,≤) with cofinite domains and images
Олег Гутік, Olha Krokhmalna · Visnyk Lvivskogo Universytetu Seriya Mekhaniko-Matematychna · 2019
Let n be a positive integer ≥ 2 and ℕ ≤ n be the n -th power of positive integers with the product order of the usual order on ℕ . In the paper we study the semigroup of injective partial monotone selfmaps of ℕ ≤ n with cofinite domains and images. We show that the group of units H ( I ) of the semigroup PO ∞ ( ℕ ≤ n ) is isomorphic to the group S n of permutations of an n -element set, and describe the subsemigroup of idempotents of PO ∞ ( ℕ ≤ n ) . Also in the case n = 3 we describe the property of elements of the semigroup PO ∞ ( ℕ ≤ 3 ) as partial bijections of the poset ℕ ≤ 3 and Green's relations on the semigroup PO ∞ ( ℕ ≤ 3 ) . In particular we show that D = J in PO ∞ ( ℕ ≤ 3 ) .