The class of completely regular semigroups $\mathcal{CR}$ endowed with the unary operation of inversion within maximal subgroups forms a variety $\mathcal{CR}$ under inclusion. The lattice of its subvarieties is denoted by $\mathcal{L(CR)}$. In previous publications, we constructed a $\cap$-subsemilattice $\Gamma$ of $\mathcal{L(CR)}$ and provided each of its members with a basis of identities.

For each of these varieties, we construct the classes of the following relations: trace, kernel, $\b B^\land$, $\b B^\lor$ as well as restrictions of these relations to $\Gamma$. A few cases of the kernel relation escape our scrutiny.

The Munn representation of an inverse semigroup $S$ provides an idempotent-separating representation by order isomorphisms between principal ideals of the semilattice $E_S$ of idempotents of $S$ and, apart from yielding important classes of inverse semigroups, it allows properties of $S$ itself to be associated with special types of semilattices. It has been generalized to various broader classes of semigroups, both regular and non-regular. In the latter case, its one-sided versions make better starting points, in the ideal situation the two combining once more into a single representation. This has been done by Gomes and Gould for left, right and two-sided Ehresmann semigroups and by the author for the broader one- and two-sided classes of $P$-Ehresmann semigroups. Here we extend this approach to the classes of DC-, RC- and DRC-semigroups introduced by Stokes. Rather than a semilattice of idempotents, we now have a set of projections that can be turned in a natural way into a projection algebra (one- or two-sided) and the representations are by order-preserving, projection-separating mappings on this projection algebra in the one-sided case, and by the product of these two in the two-sided case. As a result, these algebras of projections can be characterized axiomatically in each case. In the symmetric case, the two-sided case where the two operations on the projections are dual, we obtain a combined representation that generalizes broadly the result of Gomes and Gould and in fact represents the semigroup in an involutorial one. The involutorial case includes such semigroups as the $n \times n$ matrices over ${\mathbb C}$, for instance.

We introduce the new concept of a graph product of semigroups; this is an essentially different construction from that for monoids and groups. The second and third authors showed in a recent paper that the classes of left abundant monoids and left Fountain monoids are closed under graph products of monoids. As a corollary, via a specific embedding of a graph product of semigroups into a graph product of monoids, one can deduce the same result for semigroups. The main aim of this current paper is two-fold. First, we give a direct and relatively simple proof of the aforementioned corollary, which avoids the involved calculations in the monoid case. Second, we give the characterisation of $\mathcal{R}^*$ and $\widetilde{\mathcal{R}}$ in the graph product of semigroups, a question left open for monoids. We hope that our work here will inform a corresponding approach to the understanding of $\mathcal{R}^*$ and $\widetilde{\mathcal{R}}$ in the monoid case.

Given a bounded distributive lattice $L$ we consider the set $\widehat{L}$ of pairs $(\alpha,\beta)$ where $\alpha,\beta\in L$ and are orthogonal. We introduce a particular law of composition on $\widehat{L}$ with respect to which $\widehat{L}$ is a commutative distributive inverse monoid that satisfies the identity $x^3=x$. Moreover, the natural order in $\widehat{L}$ coincides with the cartesian order. When $L$ is pseudo-complemented, $\widehat{L}$ has maximal elements. When $L$ is a Stone algebra, every primary down-set of $\widehat{L}$ is order isomorphic to $L$, and a description of the $\mathcal H$-classes of $\widehat{L}$ is obtained in terms of its group of units which consists of the maximal elements. When $L$ is a boolean algebra, $\widehat{L}$ is semi-boolean.

The theory of congruences is an important topic in semigroup theory. There are plenty of results about congruence theory on regular semigroups. Abundant semigroups are generalizations of regular semigroups. There have been many attempts to work on congruence theory on abundant semigroups. In this paper, some results about congruences theory on abundant semigroups are summarized. We focus our survey mainly on the work of the Chinese people, of course also related to some of the work overseas.

Combinatorially factorizable cryptic restriction monoids are generalizations of combinatorially factorizable cryptic inverse monoids in the range of restriction semigroups. The structure of such semigroups is established in terms of the semidirect products of a Clifford restriction monoid and a combinatorial restriction monoid. As its applications, we give the constructions of combinatorially factorizable cryptic ample monoids and of combinatorially factorizable cryptic inverse monoids. Our results enrich and extend the related results of Petrich in [12].

The first part of the article is a survey on the relation between cluster theory and Green's equivalences in semigroup theory. Precisely, we will discuss the isomorphism classes of sub-rooted cluster algebras of a rooted cluster algebra corresponding one-by-one to the regular $\mathcal D$-classes of the semigroup consisting of partial seed endomorphisms of the initial seed. In the second part of the article, we will study a rooted cluster algebra from a Riemann surface, also we consider sub-rooted cluster algebras corresponding to the isomorphism classes of the so-called paunched surfaces.
Finally, we state our observation for rooted cluster algebras from orbifolds and give a conjectural description for them analogous to those results from the surfaces.

Every language is either the disjoint union or the intersection of two disjunctive languages. In this paper, we consider the family of i-disjunctive languages, which is a generalization of the family of disjunctive languages. Firstly, by making the usage of the unique representation for words over the same alphabet of solid codes, we consider and investigate some properties of i-disjunctive languages over an arbitrary alphabet with two letters or more; then we proceed to prove that there exists a proper f-disjunctive (resp. proper i-disjunctive) language which can be decomposed into a disjoint union of two proper i-disjunctive (resp. proper t-disjunctive) languages; finally, we introduce the ``longitudinal classification of the family of proper i-disjunctive languages.

In this paper, the concept of cubic set to ideals and closed ideals of $BG$-algebras are introduced and their related properties are studied. Relations among cubic $BG$-subalgebras, cubic ideals and cubic closed ideals of $BG$-algebras are investigated. The inverse images of cubic ideals of $BG$-algebras are defined, and how the inverse images of cubic ideals becomes cubic ideals of $BG$-algebras are studied. Also, the product of cubic $BG$-algebras are investigated.

We characterize some subdirectly irreducible bands whose structural semilattices have height 2 in terms of fundamental semilattices of semigroups.

We give a new presentation of free idempotent semigroup (band) in which the set of relations is a Gr\{o}bner-Shirshov basis. By using Composition-Diamond lemma for associative algebra, we obtain normal forms of elements of the free idempotent semigroup.

In the theory of codes, the well-known triangle conjecture was first proposed by D. Perrin and M.P. Schtzenberger in 1981. Building on the recent work [21], we further study finite maximal codes and present new findings related to triangle conjecture. This paper also provides a more concise set of results from [21] as well as amendments and corrections.