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.

Process Outcomes

Based on the aforementioned, it is now possible to allow for a discussion on how mediation strategy can account for group congruence at the sub-group level and compatibility with the mediator's goals. A mediation strategy is the roadmap approach to the regulation of the conflict, including the principles of (1) process design (process); (2) roles, views, and expectations of local and international actors (context), coordination architecture; and (3) an indication of post-agreement requirements (outcome) to enable peace (agreement) implementation.

Amenability Modulo an Ideal of Second Duals of Semigroup Algebras

The aim of this paper is to investigate the amenability modulo an ideal of Banach algebras with emphasis on applications to homological algebras. In doing so, we show that amenability modulo an ideal of A** implies amenability modulo an ideal of A. Finally, for a large class of semigroups, we prove that l1(S)** is amenable modulo Iσ** if and only if an appropriate group homomorphic image of S is finite where Iσ is the closed ideal induced by the least group congruence σ.

Reduced activity within the dorsal endogenous orienting of attention network to fearful expressions in youth with disruptive behavior disorders and psychopathic traits

Using behavioral and blood oxygen level dependent (BOLD) response indices through functional magnetic resonance imaging (fMRI), the current study investigated whether youths with disruptive behavior disorders (conduct disorder and oppositional defiant disorder) plus psychopathic traits (DBD + PT) show aberrant sensitivity to eye gaze information generally and/or whether they show particular insensitivity to eye gaze information in the context of fearful expressions. The participants were 36 children and adolescents (ages 10–17 years); 17 had DBD + PT and 19 were healthy comparison subjects. Participants performed a spatial attention paradigm where spatial attention was cued by eye gaze in faces displaying fearful, angry, or neutral affect. Eye gaze sensitivity was indexed both behaviorally and as BOLD response. There were no group differences in behavioral response: both groups showed significantly faster responses if the target was in the congruent spatial direction indicated by eye gaze. Neither group showed a Congruence × Emotion interaction; neither group showed an advantage from the displayer's emotional expression behaviorally. However, the BOLD response revealed a significant Group × Congruence × Emotion interaction. The comparison youth showed increased activity within the dorsal endogenous orienting network (superior parietal lobule and inferior parietal sulcus) for fearful congruent relative to incongruent trials relative to the youth with DBD + PT. The results are discussed with reference to current models of DBD + PT and possible treatment innovations.

Minimum Group Congruences on Eventually Regular Semigroups

An eventually regular semigroup is a semigroup in which some power of any element is regular. The minimum group congruence on an eventually regular semigroup is investigated by means of weak inverse. Furthermore, some properties of the minimum group congruence on an eventually regular semigroup are characterized.

A rank for right congruences on inverse semigroups

The S-rank (where ‘S’ abbreviates ‘sandwich’) of a right congruence ρ on a semigroup S is the Cantor-Bendixson rank of ρ in the lattice of right congruences ℛ of S with respect to a topology we call the finite type topology. If every ρ ϵ ℛ possesses S-rank, then S is ranked. It is known that every right Noetherian semigroup is ranked and every ranked inverse semigroup is weakly right Noetherian. Moreover, if S is ranked, then so is every maximal subgroup of S. We show that a Brandt semigroup 0(G, I) is ranked if and only if G is ranked and I is finite.We establish a correspondence between the lattice of congruences on a chain E, and the lattice of right congruences contained within the least group congruence on any inverse semigroup S with semilattice of idempotents E(S) ≅ E. Consequently we argue that the (inverse) bicyclic monoid B is not ranked; moreover, a ranked semigroup cannot contain a bicyclic -class. On the other hand, B is weakly right Noetherian, and possesses trivial (hence ranked) subgroups.Our notion of rank arose from considering stability properties of the theory Ts of existentially closed (right) S-sets over a right coherent monoid S. The property of right coherence guarantees that the existentially closed S-sets form an axiomatisable class. We argue that B is right coherent. As a consequence, it follows from known results that TB is a theory of B-sets that is superstable but not totally transcendental.

On joins with group congruences

Let be an arbitrary semigroup. A congruence γ on is a group congruence if /γ is a group. The set of group congruences on is non-empty since × is a group congruence. The lattice of congruences on a semigroup will be denoted by () and the set of group congruences on will be denoted by (). If () is a lattice then it is modular and γ ∨ ρ = γ ο ρ = ρ ο γ for all γ, ρ ε (). The main result is that γ ν ρ = γ ο ρ ο γ for any γ ε () and ρ ε () (whence every element of the set () is dually right modular in (). This result has appeared, for particular classes of semigroups, many times in the literature. Also γ ν ρ = γ ο ρ ο γ = ρ ο γ ο ρ for all γ, ρ ε () which is similar to the well known result for the join of congruences on a group. Furthermore, if γ ∩ ρ ε () then γ ν ρ = γ ο ρ = ρ ο γ.

EXTENSIONS OF REGULAR ORTHOGROUPS BY INVERSE SEMIGROUPS

Let V be a variety of regular orthogroups, i.e. completely regular orthodox semigroups whose band of idempotents is regular. Let S be an orthodox semigroup which is a (normal) extension of an orthogroup K from V by an inverse semigroup G, that is, there is a congruence ρ on S such that the semigroup ker ρ of all idempotent related elements of S is isomorphic to K and S/ρ≅G. It is shown that S can be embedded into an orthodox subsemigroup T of a double semidirect product A**G where A belongs to V. Moreover T itself can be chosen to be an extension of a member from V by G. If in addition ρ is a group congruence we obtain a recent result due to M.B. Szendrei [16] which says that each orthodox semigroup which is an extension of a regular orthogroup K by a group G can be embedded into a semidirect product of an orthogroup K′ by G where K′ belongs to the variety of orthogroups generated by K.