The algebraic study of braces is crucial to classify set-theoretical solutions of the Yang-Baxter equation (YBE). In this survey we introduce and study the class of {\it Dedekind} braces, or braces in which every subbrace is an ideal, and show important algebraic properties and structural results. As a consequence, applications to the classification problem of set-theoretical solutions of the YBE naturally emerge.

 The aim of this paper is to investigate the structure of semilattices of monoids via structural homomorphisms, with special attention to semilattices of unipotent (one idempotent) monoids. This is an extension of the well-known construction of Clifford semigroups (semilattices of groups). It is similar to the construction of a right abundant semigroup with central idempotents, characterized by Fountain as a strong semilattice of left cancellative monoids.

 The aim of this paper is to consider the lattice $\mathfrak{F}(S)$ of full subsemigroups of a (left; right) restriction semigroup $S$. It is verified that $\mathfrak{F}(S)$ is a complete lattice when $S$ is a restriction semigroup. The structure of $\mathfrak{F}(S)$ is established. In particular, we determine when $\mathfrak{F}(S)$ is a chain.

 The subsets of $[n]=\{ 1, 2, \ldots , n \}$ form a poset $B_n$ with respect to inclusion. A family $\mathcal{F} \subset 2^{[n]}$ of subsets can be considered as a subposet. Let $H$ be a given ``small" poset, we say that $\mathcal{F}$ is $H$-free if $\mathcal{F}$ contains no subposet isomorphic to $H$. A family is intersecting if the intersection of any pair of its members is non-empty. The poset $V$ has 3 elements, $x, y, z$ satisfying the relations $x < y, z$ while the elements of the poset $\Lambda$ satisfy the relations $ x, y < z$. We determine the exact maximum of the size of an intersecting family of sets containing neither a $V$ nor a $\Lambda$.

 In this paper, we study the influence of the order of finite groups and their element orders on structure of the finite groups. We give a new characterization of all sporadic simple groups and alternating groups by using their orders and prime graph. Let $G$ be a group. We show that if $T$ is a sporadic simple group, $|G|=|T|$, $G$ and $T$ have common $NC$ primes, then $G\cong T$; if $T$ is an alternating group $A_n$($n \geq 5$), $|G|=|T|$, $G$ has the same arithmetic complex as $T$, then $G\cong T$.

 Cross-connection is a categorical duality that enables us to obtain a representation of abstract mathematical structures. In 1994, K.S.S. Nambooripad described the structure of arbitrary regular semigroups using normal categories and their cross-connection. In this paper, we describe the cross-connection representation of right zero semigroups, a special class of regular semigroups. Further, it is also noted that there is precisely one cross-connection between the principal left and right ideal categories $ \mathscr L(S)$ band $\mathscr R(S)$ of a right zero semigroup $S$.

 We introduce the notion of {\it WRBS generalised categories}. Our purpose is to describe a class of semigroups which we name {\it weakly $E$-regular semigroups}. Here $E$ is a regular biordered set. Weakly $E$-regular semigroups are analogues of regular semigroups and abundant semigroups with a regular biordered set of idempotents, where the relations $\widetilde{\mathcal{R}}_E$ and $\widetilde{\mathcal{R}}_E$ play the role that $\ar$ and $\el$ take in the regular case. We show that the category of weakly $E$-regular semigroups and admissible morphisms is isomorphic to the category of WRBS generalised categories over regular biordered sets and pseudo-functors. In addition, we show that $T^\ast(E)$ is an IC-RBS category, where $T^{\ast}(E)$ is the set of all $\omega$-isomorphisms of a regular biordered set $E$. 

 A $\ast$-band is a triple $(B, \cdot, ^\circ)$ in which $(B, \cdot)$ is an idempotent semigroup and $\circ: S\longrightarrow S$ is a map satisfying the axioms: $x x^\circ x=x,\, x^{\circ\circ}=x,\, (xy)^\circ=y^\circ x^\circ.$ A $\ast$-band is called a normal $\ast$-band if it also satisfies the axiom $xyzw=xzyw$. For a given $\ast$-band $(B, \cdot, ^\circ)$, one can obtain its Hall semigroup $W_B$ and Jones semigroup $T_B$. In this note, we show that $W_B\cong T_B$ for any normal $\ast$-band $(B, \cdot, ^\circ)$. This partially solves a problem raised by Shoufeng Wang and Kar Ping Shum.

 The category of principal left [or right] $\ast$-ideals of an abundant semigroup has been abstractly characterised as a balanced category. Cross connection is a relation connecting two balanced categories $\categc$ and $\categd$ so that $\categc$ is isomorphic to the balanced category of principal left $\ast$-ideals of an abundant semigroup and $\categd$ is isomorphic to the balanced category of principal right $\ast$-ideals of the same abundant semigroup. Here we describe all categories $\categd$ for which there is a cross connection of $\categd$ with one object balanced category $\categc$. We also describe all cross connections of them and the cross connection semigroups. The results of this paper generalized similar results for normal categories obtained recently by A.R. Rajan. 

 Locality semigroup has been proposed recently as one of the basic locality algebraic structures, which are studied extensively in quantum field theory. In this paper, we introduce the notion of a refined locality semigroup, and show that the path locality semigroup of a quiver is the free refined locality semigroup. We also explore the relationships among locality semigroups, partial semigroups and path locality semigroups, concluding that the path locality semigroup is a proper subclass of the intersection of locality semigroups and partial semigroups. Furthermore, one can obtain semigroups from refined locality semigroups by Brandt locality semigroups.