We define a new semigroup construction $P_{\lambda}^{\mathcal{F}}$ which generalizes the polycyclic monoid and discuss its algebraic properties.

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We describe endomorphisms of the bicyclic monoid $\boldsymbol{B}_{\omega}$ as a monoid and as a semigroup and endomorphisms of the extended bicyclic semigroup $\boldsymbol{B}_{\mathbb{Z}}$ a. It is proved that the semigroup $\mathrm{\mathbf{End}}(\boldsymbol{B}_{\omega})$ of endomorphisms of the bicyclic semigroup $\boldsymbol{B}_{\omega}$ is isomorphic to the semidirect product $(\omega,+)\rtimes_\varphi(\omega,*)$ and the semigroup $\mathrm{\mathbf{End}}(\boldsymbol{B}_{\mathbb{Z}})$ of endomorphisms of the extended bicyclic semigroup $\boldsymbol{B}_{\mathbb{Z}}$ is isomorphic to the semidirect product $\mathbb{Z}(+)\rtimes_\varphi(\omega,*)$.

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We discuss classical results of Tamura and Petrich about the binary quasiorder on a semigroup. This quasiorder induces the smallest semilattice congruence, that divides the semigroup into semilattice-indecomposable subsemigroups.

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A semigroup X is called E-separated if homomorphisms to semilattices separate points of X. We present several characterizations of E-separated semigroups and discuss their applications to categorically closed semigroups.

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A semigroup X is called E-separated if homomorphisms to semilattices separate points of X. We present several characterizations of E-separated semigroups and discuss their applications to categorically closed semigroups.

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We study automorphisms of the semigroup ℬ_ω^ℱ with the family ℱ of inductive nonempty subsets of ω. The group Aut(ℬ_ω^ℱ) of automorphisms of the semigroup ℬ_ω^ℱ is described. In particular, we prove that if the family ℱ is infinite then Aut(ℬ_ω^ℱ) is isomorphic to the additive group integers, and in the case when the family ℱ contains k+1 inductive nonempty subsets of ω, especially ℱ = {[0), …, [k)}, then the group Aut(ℬ_ω^ℱ) is isomorphic to the group ⟨ x,y | xy=yx, y²=x^k ⟩.

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We study the semigroup ℬ_ω^ℱ, which is introduced in [O. Gutik and M. Mykhalenych, On some generalization of the bicyclic monoid, Visnyk Lviv. Univ. Ser. Mech.-Mat. 90 (2020), 5-19 (in Ukrainian)], in the case when the family ℱ_n generated by the set {0,1,...,n}. We show that the Green relations 𝒟 and 𝒥 coincide in ℬ_ω^ℱ_n, the semigroup ℬ_ω^ℱ_n is isomorphic to the semigroup of partial order preserving convex bijections of ω of the rank ≤ n+1, and ℬ_ω^ℱ admits only Rees congruences.

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