Bulletin of the Iranian Mathematical Society

For a non-empty set X denote the full transformation semigroup on X by T(X) and suppose that $$\sigma $$ is an equivalence relation on X. For every $$f\in T(X)$$ , the kernel of f is defined to be $$\ker f =\{(x, y)\in X\times X\mid f(x) = f(y)\}$$ . Evidently, $$E(X, \sigma )=\{f\in T(X) \mid \sigma \subseteq \ker f\}$$ is a subsemigroup of T(X). Also, the subset $$RE(X, \sigma )$$ of $$E(X, \sigma )$$ consisting of regular elements is a subsemigroup. Partition of a semigroup by Green’s $$*$$ -relations was first introduced by Fountain in 1979 and the Green’s $$\sim $$ -relations (with respect to a non-empty subset U of the set of idempotents) as a new method of partition were introduced by Lawson (J Algebra 141(2):422–462, 1991). In this paper, we intend to present certain characterizations of these two sets of Green’s relations of the semigroup $$E(X, \sigma )$$ . This investigation proves that the semigroup $$E(X, \sigma )$$ is always a right Ehresmann semigroup. Finally, we prove that $$RE(X, \sigma )$$ is an orthodox semigroup if and only if the set X consists of at most two $$\sigma $$ -classes.

The paper studies partially observed optimal control problems of general McKean–Vlasov differential equations, in which the coefficients depend on the state of the solution process as well as of its law and the control variable. By applying Girsanov’s theorem with a standard variational technique, we establish a stochastic maximum principle on the assumption that the control domain is convex. As an application, partially observed linear-quadratic control problem is discussed.

The aim of this note is to use some structural properties of quasi-commutative semigroups to get information on their ideals.

Let p be an analytic function defined on the open unit disc $$\mathbb {D}$$ with $$p(0)=1$$ and $$0< \alpha \le 1$$ . The conditions on complex valued functions C, D, and E are obtained for p to be subordinate to $$((1+z)/(1-z))^{\alpha }$$ when $$C(z) z^{2}p''(z)+D(z)zp'(z) + E(z)p(z)=0$$ . Sufficient conditions for confluent (Kummer) hypergeometric function and generalized and normalized Bessel function of the first kind of complex order to be subordinate to $$((1+z)/(1-z))^{\alpha }$$ are obtained as applications. The conditions on $$\alpha $$ and $$\beta $$ are derived for p to be subordinate to $$((1+z)/(1-z))^{\alpha }$$ when $$1+\beta zp'(z)/p^{n}(z)$$ with $$n=1,2$$ is subordinate to $$1+4z/3+2z^{2}/3=:\varphi _{CAR}(z)$$ . Similar problems were investigated for $${{\,\mathrm{Re}\,}}p(z)>0$$ when the function $$p(z)+\beta zp'(z)/p^{n}(z)$$ with $$n=0,2$$ is subordinate to $$\varphi _{CAR}(z)$$ . The condition on $$\beta $$ is determined for p to be subordinate to $$((1+z)/(1-z))^{\alpha }$$ when $$p(z)+\beta zp'(z)/p^{n}(z)$$ with $$n=0,1,2$$ is subordinate to $$((1+z)/(1-z))^{\alpha }$$ .

In this paper, we first characterize when a semigroup has completely 0-simple semigroup as its principal factors. Let R be a commutative ring with an identity, and let S be a locally inverse semigroup with the set of idempotents locally pseudofinite. Assume that the principal factors of S are all completely 0-simple. Then, we prove that the contracted semigroup algebra $$R_0[S]$$ is $$\pi $$ -semisimple if and only if the contracted semigroup algebras of all the principal factors of S are $$\pi $$ -semisimple. Examples are provided to illustrate that the locally pseudofinite condition on the idempotent set of S cannot be removed. Notice that we extend the corresponding results on finite locally inverse semigroups.

Recently, Phuong and Thin (Ukrain Math J 67(7), 2015) proved some fundamental theorems for holomorphic curves on the annuli with target being fixed hyperplanes. In this paper, we consider the same problems in the case of target being fixed hypersurfaces.