Abstract:
Let $S$ be a semigroup and $T$ be a subsemigroup of finite index in $S$ (that is, the set $S\setminus T$ is finite). The subsemigroup $T$ is also called a large subsemigroup of $S$. It is well known that if $T$ has a finite complete rewriting system then so does $S$. In this paper, we will prove the converse, that is, if $S$ has a finite complete rewriting system then so does $T$. Our proof is purely combinatorial and also constructive.

Abstract:
The class of finitely presented monoids defined by homogeneous (length-preserving) relations is considered. The properties of admitting a finite complete rewriting system, having finite derivation type, being automatic, and being biautomatic, are investigated for monoids in this class. The first main result shows that for any possible combination of these properties and their negations there is a homoegenous monoid with exactly this combination of properties. We then extend this result to show that the same statement holds even if one restricts attention to the class of $n$-ary multihomogeneous monoids (meaning every side of every relation has fixed length $n$, and all relations are also content preserving).

Abstract:
Every tame, prime and alternating knot is equivalent to a tame, prime and alternating knot in regular position, with a common projection. In this work, we show that the Dehn presentation of the knot group of a tame, prime, alternating knot, with a regular and common projection has a finite and complete rewriting system. Although there are rules in the rewriting system with left-hand side a generator and which increase the length of the words we show that the system is terminating.

Abstract:
We prove that every monoid $\mathrm{Mon}\langle a,b:a^{\alpha}b^{\beta}a^{\gamma}b^{\delta}=b\rangle$ admits a finite complete rewriting system. Furthermore we prove that $\mathrm{Mon}\langle a,b:ab^2a^2b^2=b\rangle$ is non-hopfian, providing an example of a finitely presented non-residually finite monoid with linear Dehn function.

Abstract:
Let p be an odd prime. We show that the classification of p-divisible groups by Breuil windows and the classification of finite flat group schemes of p-power order by Breuil modules hold over any complete regular local ring with perfect residue field of characteristic p. We use a formalism of frames and windows with an abstract deformation theory that applies to Breuil windows.

Abstract:
As we know if D is a complete X-semilattice of unions then semigroup Bx(D)possesses a right unit iff D is an XI-semilattice of unions. The investigation of those a-idempotent and regular elements of semigroups B_{x}(D) requires an investigation of XI-subsemilattices of semilattice D for which V(D,a)=Q∈∑_{2}(X,8) . Because the semilattice Q of the class ∑_{2}(X,8) are not always XI -semilattices, there is a need of full description for those idempotent and regular elements when V(D,a)=Q . For the case where X is a finite set we derive formulas by calculating the numbers of such regular elements and right units for which V(D,a)=Q .

Abstract:
The aim of this note is to prove that monoids $\mathrm{Mon}\langle a,b:aUb=b\rangle$, with $aUb$ of relative length 6, admit finite complete rewriting systems. This is some advance in the understanding the long-standing open problem whether the word problem for one-relator monoids is soluble.

Abstract:
This paper constructs presentations via finite complete rewriting systems for Plactic monoids of types $A_n$, $B_n$, $C_n$, $D_n$, and $G_2$, using a unified proof strategy that depends on Kashiwara's crystal bases and analogies of Young tableaux, and on Lecouvey's presentations for these monoids. As corollaries, we deduce that Plactic monoids of these types have finite derivation type and satisfy the homological finiteness properties left and right $\mathrm{FP}_\infty$. These rewriting systems are then applied to show that Plactic monoids of these types are biautomatic.

Abstract:
Combining the the results of A.R. Meyer and L.J. Stockmeyer "The Equivalence Problem for Regular Expressions with Squaring Requires Exponential Space", and K.S. Booth "Isomorphism testing for graphs, semigroups, and finite automata are polynomiamlly equivalent problems" shows that graph isomorphism is PSPACE-complete.

Abstract:
In this paper, we take Q_{16} subsemilattice of D and we will calculate the number of right unit, idempotent and regular elements α of B_{X} (Q_{16}) satisfied that V (D, α) = Q_{16} for a finite set X. Also we will give a formula for calculate idempotent and regular elements of B_{X} (Q) defined by an X-semilattice of unions D.