On inverse submonoids of the monoid of almost monotone injective co-finite partial selfmaps of positive integers

In this paper we study submonoids of the monoid $\mathscr{I}_{\infty}^{\,\Rsh\!\!\nearrow}(\mathbb{N})$ of almost monotone injective co-finite partial selfmaps of positive integers $\mathbb{N}$. Let $\mathscr{I}_{\infty}^{\nearrow}(\mathbb{N})$ be a submonoid of $\mathscr{I}_{\infty}^{\,\Rsh\!\!\nearrow}(\mathbb{N})$ which consists of cofinite monotone partial bijections of $\mathbb{N}$ and $\mathscr{C}_{\mathbb{N}}$ be a subsemigroup of $\mathscr{I}_{\infty}^{\,\Rsh\!\!\nearrow}(\mathbb{N})$ which is generated by the partial shift $n\mapsto n+1$ and its inverse partial map. We show that every automorphism of a full inverse subsemigroup of $\mathscr{I}_{\infty}^{\!\nearrow}(\mathbb{N})$ which contains the semigroup $\mathscr{C}_{\mathbb{N}}$ is the identity map. We construct a submonoid $\mathbf{I}\mathbb{N}_{\infty}^{[\underline{1}]}$ of $\mathscr{I}_{\infty}^{\,\Rsh\!\!\nearrow}(\mathbb{N})$ with the following property: if $S$ is an inverse submonoid of $\mathscr{I}_{\infty}^{\,\Rsh\!\!\nearrow}(\mathbb{N})$ such that $S$ contains $\mathbf{I}\mathbb{N}_{\infty}^{[\underline{1}]}$ as a submonoid, then every non-identity congruence $\mathfrak{C}$ on $S$ is a group congruence. We show that if $S$ is an inverse submonoid of $\mathscr{I}_{\infty}^{\,\Rsh\!\!\nearrow}(\mathbb{N})$ such that $S$ contains $\mathscr{C}_{\mathbb{N}}$ as a submonoid then $S$ is simple and the quotient semigroup $S/\mathfrak{C}_{\mathbf{mg}}$, where $\mathfrak{C}_{\mathbf{mg}}$ is the minimum group congruence on $S$, is isomorphic to the additive group of integers. Also, we study topologizations of inverse submonoids of $\mathscr{I}_{\infty}^{\,\Rsh\!\!\nearrow}(\mathbb{N})$ which contain $\mathscr{C}_{\mathbb{N}}$ and embeddings of such semigroups into compact-like topological semigroups.

On the monoid of cofinite partial isometries of $\qq{N}^n$ with the usual metric

In this paper we study the structure of the monoid Iℕn ∞ of cofinite partial isometries of the n-th power of the set of positive integers ℕ with the usual metric for a positive integer n > 2. We describe the group of units and the subset of idempotents of the semigroup Iℕn ∞, the natural partial order and Green's relations on Iℕn ∞. In particular we show that the quotient semigroup Iℕn ∞/Cmg, where Cmg is the minimum group congruence on Iℕn ∞, is isomorphic to the symmetric group Sn and D = J in Iℕn ∞. Also, we prove that for any integer n ≥2 the semigroup Iℕn ∞ is isomorphic to the semidirect product Sn ×h(P∞(Nn); U) of the free semilattice with the unit (P∞(Nn); U) by the symmetric group Sn.

On the special semigroup “at infinity”

This paper presents an important new technique for studying a particular compact semigroup, N∪{∞}, the one-point compactification of positive integers with usual addition, which is an important semigroup. Indeed, the semigroup N ∪ {∞} is constructed as the quotient semigroup of a particular compact right topological semigroup. In the study of such a semigroup, a major role is played by the substructures called standard oids. For instance, some of the already known results on the structure of N ∪ {∞} are obtained as immediate consequences.

Nilpotents and congruences on semigroups of transformations with fixed rank

In 1988, Howie and Marques-Smith studied Pm, a Rees quotient semigroup of transformations associated with a regular cardinal m, and described the elements which can be written as a product of nilpotents in Pm. In 1981, Marques proved that if Δm denotes the Malcev congruence on Pm, then Pm/Δm is congruence-free for any infinite m. In this paper, we describe the products of nilpotents in Pm when m is nonregular, and determine all the congruences on Pm when m is an arbitrary infinite cardinal. We also investigate when a nilpotent is a product of idempotents.

Commutative Non-Singular Semigroups

It is well known (see [5]) that the maximal right quotient ring of a ring R is (von Neumann) regular if and only if R is (right) non-singular (every large right ideal is dense). In [8] it was shown that for a semigroup S, the regularity of Q(S), the maximal right quotient semigroup [7], is independent of the non-singularity of S. Nevertheless, right non-singular semigroups form an important class of semigroups.

The Singular Congruence and the Maximal Quotient Semigroup

It is a well known result (see [4, p. 108]) that if R is a ring and Q(R) its maximal right quotient ring, then Q(R) is (von Neumann) regular if and only if every large right ideal of R is dense. This condition is equivalent to saying that the singular ideal of R is zero. In this note we show that the condition loses its magic in the theory of semigroups.

The quotient semigroup of a semigroup that is a semilattice of groups

Let Q(S) denote the maximal right quotient semigroup of the semigroup S as defined in [4]. In this paper, we initiate a study of Q(S) when S is a semilattice of groups. A structure theorem for such semigroups is given by Theorem 4.11 of [2].