Semigroups of transformations whose characters belong to a given semigroup

Let $X$ be a nonempty set and $\mathcal{P}=\{X_i\colon i\in I\}$ a partition of $X$. Denote by $T(X)$ the full transformation semigroup on $X$, and $T(X, \mathcal{P})$ the subsemigroup of $T(X)$ consisting of all transformations that preserve $\mathcal{P}$. For every subsemigroup $\mathbb{S}(I)$ of $T(I)$, let $T_{\mathbb{S}(I)}(X,\mathcal{P})$ be the semigroup of all transformations $f\in T(X, \mathcal{P})$ such that $\chi^{(f)}\in \mathbb{S}(I)$, where $\chi^{(f)}\in T(I)$ defined by $i\chi^{(f)}=j$ whenever $X_if\subseteq X_j$. We describe regular and idempotent elements in $T_{\mathbb{S}(I)}(X,\mathcal{P})$, and determine when $T_{\mathbb{S}(I)}(X,\mathcal{P})$ is a regular semigroup [inverse semigroup]. With the assumption that $\mathbb{S}(I)$ contains the identity, we characterize Green's relations on $T_{\mathbb{S}(I)}(X,\mathcal{P})$, describe unit-regular elements in $T_{\mathbb{S}(I)}(X,\mathcal{P})$, and determine when $T_{\mathbb{S}(I)}(X,\mathcal{P})$ is a unit-regular semigroup. We apply these general results to obtain more concrete results for $T(X,\mathcal{P})$.Comment: 18 page

Unit-regular elements in restrictive semigroups of transformations and linear operators

Let $T(X)$ be the full transformation semigroup on a set $X$ and let $L(V)$ be the semigroup under composition of all linear operators on a vector space $V$ over a field. For a nonempty subset $Y$ of $X$ and a subspace $W$ of $V$, we consider the restrictive semigroups $\overline{T}(X, Y) = \{f\in T(X)\mid Yf \subseteq Y\}$ and $\overline{L}(V, W) = \{f\in L(V)\mid Wf \subseteq W\}$ under composition. We characterize unit-regular elements in $\overline{T}(X, Y)$ and $\overline{L}(V, W)$. Utilizing these, we characterize unit-regularity of $\overline{T}(X, Y)$ and $\overline{L}(V, W)$. We prove that $f\in L(V)$ is unit-regular if and only if nullity$(f) = {\rm corank}(f)$. A transformation semigroup is called semi-balanced if all its elements are semi-balanced. We determine a necessary and sufficient condition for $\overline{T}(X, Y)$ and $L(V)$ to be semi-balanced.Comment: 12 page

The Holonomy Decomposition of some Circular Semi-Flower Automata

Using holonomy decomposition, the absence of certain types of cycles in automata has been characterized. In the direction of studying the structure of automata with cycles, this paper focuses on a special class of semi-flower automata and establish the holonomy decomposition of certain circular semiflower automata. In particular, we show that the transformation monoid of a circular semi-flower automaton with at most two bpis divides a wreath produt of cyclic transformation groups with adjoined constant functions

The experience with repetitive transcranial magnetic stimulation as add-on treatment in the elderly with depression: A preliminary report

Background: Elderly depression is a fairly common and often difficult to treat condition. Elderly patients also often have comorbid medical conditions that preclude the use of other somatic treatment modalities. Repetitive transcranial magnetic stimulation (rTMS) is a treatment methodology that is approved to be used in depression and is supposed to have fewer side-effects. This paper describes the experience of a recently started rTMS service in a tertiary hospital in North India with referred elderly patients suffering from depression. Methods: Results of rTMS therapy administered to 7 elderly patients who were referred during this period are described. Results: Only one patient with bipolar depression perceived significant benefit from rTMS. Three patients complained of mild and transient side-effects, and one patient discontinued treatment due to his medical condition (unrelated to rTMS). Conclusions: rTMS seems to be safe and well-tolerated in this population. However, further experience is needed before commenting definitely on effectiveness of this treatment modality