The Algebraic Theory of Semigroups, Vol. I. By A. H. Clifford and G. B. Preston. Pp. xv + 224. $10.60. 1961. (American Mathematical Society, Providence.)

We study algebraic structure of the $\lambda$-polycyclic monoid $P_{\lambda}$ and its topologizations. We show that the $\lambda$-polycyclic monoid for an infinite cardinal $\lambda\geqslant 2$ has similar algebraic properties so has the polycyclic monoid $P_n$ with finitely many $n\geqslant 2$ generators. In particular we prove that for every infinite cardinal $\lambda$ the polycyclic monoid $P_{\lambda}$ is a congruence-free combinatorial $0$-bisimple $0$-$E$-unitary inverse semigroup. Also we show that every non-zero element $x$ is an isolated point in $(P_{\lambda},\tau)$ for every Hausdorff topology $\tau$ on $P_{\lambda}$, such that $(P_{\lambda},\tau)$ is a semitopological semigroup, and every locally compact Hausdorff semigroup topology on $P_\lambda$ is discrete. The last statement extends results of the paper [33] obtaining for topological inverse graph semigroups. We describe all feebly compact topologies $\tau$ on $P_{\lambda}$ such that $\left(P_{\lambda},\tau\right)$ is a semitopological semigroup and its Bohr compactification as a topological semigroup. We prove that for every cardinal $\lambda\geqslant 2$ any continuous homomorphism from a topological semigroup $P_\lambda$ into an arbitrary countably compact topological semigroup is annihilating and there exists no a Hausdorff feebly compact topological semigroup which contains $P_{\lambda}$ as a dense subsemigroup.

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On a semitopological polycyclic monoid

A semigroup $T$ is called a  Taimanov semigroup if $T$ contains two distinct elements $0,\infty$ such that $xy=\infty$ for any distinct points $x,y\in T\setminus\{0,\infty\}$ and $xy=0$ in all other cases.  We prove that any Taimanov semigroup $T$ has the following topological properties: (i) each $T_1$-topology with continuous shifts on $T$ is discrete; (ii) $T$ is closed in each $T_1$-topological semigroup containing $T$ as a subsemigroup; (iii) every non-isomorphic homomorphic image $Z$ of $T$ is a zero-semigroup and hence $Z$ is a topological semigroup in any topology on $Z$.

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Topological properties of Taimanov semigroups

In this paper we study the semigroup I∞ λ of injective partial selfmaps almost everywhere the identity of a set of infinite cardinality λ. We describe the Green relations on I∞ λ , all (two-sided) ideals and all congruences of the semigroup I∞ λ . We prove that every Hausdorff hereditary Baire topology τ on I∞ ω such that (I ∞ ω , τ) is a semitopological semigroup is discrete and describe the closure of the discrete semigroup I∞ λ in a topological semigroup. Also we show that for an infinite cardinal λ the discrete semigroup I∞ λ does not embed into a compact topological semigroup and construct two non-discrete Hausdorff topologies turning I∞ λ into a topological inverse semigroup.

On monoids of injective partial selfmaps almost everywhere the identity

We describe the structure of Hausdorff locally compact semitopological $0$-bisimple inverse $\omega$-semigroups with compact maximal subgroups. In particular, we show that a Hausdorff locally compact semitopological $0$-bisimple inverse $\omega$-semigroup with a compact maximal subgroup is either compact or topologically isomorphic to the topological sum of its $\mathscr{H}$-classes. We describe the structure of Hausdorff locally compact semitopological $0$-bisimple inverse $\omega$-semigroups with a monothetic maximal subgroups. In particular we prove the dichotomy for $T_1$ locally compact semitopological Reilly semigroup $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^0,\tau\right)$ with adjoined zero and with a non-annihilating homomorphism $\theta\colon \mathbb{Z}_{+}\to \mathbb{Z}_{+}$: $\left(\textbf{B}(\mathbb{Z}_{+},\theta)^0,\tau\right)$ is either compact or discrete. At the end we discuss on the remainder under the closure of the discrete Reilly semigroup $\textbf{B}(\mathbb{Z}_{+},\theta)^0$ in a semitopological semigroup.

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On locally compact semitopological $0$-bisimple inverse $\omega$-semigroups

In this paper we study the semigroup $I_\infty^\dnearrow(N)$ of partial co-finite almost monotone bijective transformations of the set of positive integers $\mathbb{N}$. We show that the semigroup $I_\infty^\dnearrow(N)$ has algebraic properties similar to the bicyclic semigroup: it is bisimple and all of its non-trivial group homomorphisms are either isomorphisms or group homomorphisms. Also we prove that every Baire topology $\tau$ on $I_\infty^\dnearrow(N)$ such that $(I_\infty^\dnearrow(N),\tau)$ is a semitopological semigroup is discrete, describe the closure of $(I_\infty^\dnearrow(N),\tau)$ in a topological semigroup and construct non-discrete Hausdorff semigroup topologies on $I_\infty^\dnearrow(N)$.

Topological monoids of almost monotone injective co-finite partial selfmaps of positive integers

In this paper we study the semigroup $\mathscr{I}^{\infty}_\lambda$ of injective partial selfmaps almost everywhere the identity of a set of infinite cardinality $\lambda$ We describe the Green relations on $\mathscr{I}^{\infty}_\lambda$, all (two-sided) ideals and all congruences of the semigroup $\mathscr{I}^{\infty}_\lambda$ We prove that every Hausdorff hereditary Baire topology $\tau$ on $\mathscr{I}^{\infty}_\omega$ such that $(\mathscr{I}^{\infty}_\omega,\tau)$ is a semitopological semigroup is discrete and describe the closure of the discrete semigroup $\mathscr{I}^{\infty}_\lambda$ in a topological semigroup Also we show that for an infinite cardinal $\lambda$ the discrete semigroup $\mathscr{I}^{\infty}_\lambda$ does not embed into a compact topological semigroup and construct two non-discrete Hausdorff topologies turning $\mathscr{I}^{\infty}_\lambda$ into a topological inverse semigroup

In the paper we study the semigroup I 1 of almost identity injective partial selfmaps of the set of cardinality . We describe the Green relations on I 1 , all (two-sided) ideals and all congruences of the semigroup I 1 . We prove that every Hausdor hereditary Baire topology on I 1 such that (I 1 ; ) is a semitopological semigroup is discrete and describe the closure of the discrete semigroup I 1 in a topological semigroup. Also we show that the discrete semigroup I 1 does not embed into a compact topological semigroup and construct two non-discrete Hausdor topologies turning I 1 into a topological inverse semigroup.

Ivan Chuchman

Papers

On monoids of injective partial selfmaps almost everywhere the identity

Topological monoids of almost monotone injective co-finite partial selfmaps of positive integers

On monoids of injective partial selfmaps almost everywhere the identity

On monoids of almost identity injective partial selfmaps