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Semi-regular sequences over $\mathbb{F}_2$ are sequences of homogeneous elements of the algebra $ B^{(n)}=\mathbb{F}_2[X_1,...,X_n]/(X_1^2,...,X_n^2) $, which have as few relations between them as possible. They were introduced in order to…

Let $G$ be a countable cancellative amenable semigroup and let $(F_n)$ be a (left) F{\o}lner sequence in $G$. We introduce the notion of an $(F_n)$-normal element of $\{0,1\}^G$. When $G$ = $(\mathbb N,+)$ and $F_n = \{1,2,...,n\}$, the…

Let $S$ be an algebraic semigroup (not necessarily linear) defined over a field $F$. We show that there exists a positive integer $n$ such that $x^n$ belongs to a subgroup of $S(F)$ for any $x \in S(F)$. In particular, the semigroup $S(F)$…

By a classical theorem of Jordan, every faithful transitive action of a nontrivial finite group has a derangement (an element with no fixed points). The existence of derangements with additional properties has attracted much attention,…

Let X be a coherent configuration associated with a transitive group G. In terms of the intersection numbers of X, a necessary condition for the point stabilizer of G to be a TI-subgroup, is established. Furthermore, under this condition, X…

A semigroup (dynamical system) generated by $C^{1+\alpha}$-contracting mappings is considered. We call a such semigroup regular if the maximum $K$ of the conformal dilatations of generators, the maximum $l$ of the norms of the derivatives…

A set of vertices $S$ in a simple isolate-free graph $G$ is a semi-total dominating set of $G$ if it is a dominating set of $G$ and every vertex of $S$ is within distance 2 or less with another vertex of $S$. The semi-total domination…

A finite group $G$ is called an F-group if for every $x, y \in G \setminus Z(G)$, $C(x) \leq C(y)$ implies that $C(x) = C(y)$. On the otherhand, two elements of a group are said to be $z$-equivalent or in the same $z$-class if their…

A theorem of Dolfi, Herzog, Kaplan, and Lev \cite[Thm.~C]{DHKL} asserts that in a finite group with trivial Fitting subgroup, the size of the soluble residual of the group is bounded from below by a certain power of the group order, and…

Groups with a topology that is in consistent one way or another with the algebraic structure are considered. Classical groups with a topology are topological, paratopological, semitopological, and quasitopological groups. We also study…

A numerical semigroup is a submonoid of ${\mathbb Z}_{\ge 0}$ whose complement in ${\mathbb Z}_{\ge 0}$ is finite. For any set of positive integers $a,b,c$, the numerical semigroup $S(a,b,c)$ formed by the set of solutions of the inequality…

The note contains a few results related to separation axioms and automatic continuity of operations in compact-like semitopological groups. In particular, is presented a semiregular semitopological group $G$ which is not $T_3$. We show that…

We prove a natural generalization of Szep's conjecture. Given an almost simple group $G$ with socle not isomorphic to an orthogonal group having Witt defect zero, we classify all possible group elements $x,y\in G\setminus\{1\}$ with $G={\bf…

The reconstruction theorem and the multilevel Schauder estimate have central roles in the analytic theory of regularity structures [17]. Inspired by [26], we provide elementary proofs for them by using the semigroup of operators.…

In this paper we introduce a quotient structure on topological ternary semigroup by defining a congruence suitably. We have found conditions under which this quotient structure becomes a topological ternary semigroup. We have also obtained…

Let $R$ be an algebra over a ring $\Bbbk$, $T$ an $R$-algebra, $M$ a finitely generated projective $R$-module, and $N$ a $T$-module. Let $G$ be a linearly reductive group scheme over $\Bbbk$ equipped with a representation…

Let $H$ be a subgroup of a finite group $G$. We say that $H$ satisfies partial $\Pi$-property in $G$ if there exists a chief series $\mathit{\Gamma}_G:1=G_0<G_1<\cdots<G_n=G$ of $G$ such that for every $G$-chief factor $G_i/G_{i-1}$ ($1\leq…

Let G be a semigroup of rational functions of degree at least two where the semigroup operation is composition of functions. We prove that the largest open subset of the Riemann sphere on which the semigroup G is normal and is completely…

Let $\mathcal{X}$ be a separable Hilbert space with norm $\|\cdot\|$ and let $T>0$. Let $Q$ be a linear, self-adjoint, positive, trace class operator on $\mathcal{X}$, let $F:\mathcal{X}\rightarrow \mathcal{X}$ be a (smooth enough) function…

We keep the definition of intra-regularity (left regularity) of $po$-$\Gamma$-semigroups introduced in arXiv: 1511.00679 which is absolutely necessary for the investigation. Being able to describe the form of the elements of the principal…