Combinatorial results for certain semigroups of order-decreasing partial isometries of a finite chain

Presentations for subsemigroups of $PD_n$

Let $[n]=\{1,\ldots,n\}$ be the $n$-chain. We give presentations for the following transformation semigroups: the semigroup of full order-decreasing mappings of $[n]$, the semigroup of partial one-to-one order-decreasing mappings of $[n]$, the semigroup of full order-preserving and order-decreasing mappings of $[n]$, the semigroup of partial one-to-one order-preserving and order-decreasing mappings of $[n]$, and the semigroup of partial order-preserving and order-decreasing mappings of $[n]$.

On certain Semigroup of Order-decreasing Full Contraction Mappings of a Finite Chain

Let $\mathcal{CT}_n$ be the semigroup of full contraction mappings on $[n]=\{1,2,\ldots,n\}$, and let $\mathcal{OCT}_n$ and $\mathcal{ODCT}_n$ be the subsemigroups consisting of all order-preserving full contraction and subsemigroup of order-decreasing and order-preserving full contraction mappings, respectively. In this paper, we show that the semigroup $\mathcal{ODCT}_n$ is left adequate. We further study the rank properties and as well obtain the rank of the semigroup, $\mathcal{ODCT}_n$. Moreover, we obtain a characterization of natural partial order for the semigroup $\mathcal{OCT}_n$ and…

On the monoid of cofinite partial isometries of $\mathbb{N}^n$ with the usual metric

In this paper we study the structure of the monoid $\mathbf{I}\mathbb{N}_{\infty}^n$ of cofinite partial isometries of the $n$-th power of the set of positive integers $\mathbb{N}$ with the usual metric for a positive integer $n\geqslant 2$. We describe the elements of the monoid $\mathbf{I}\mathbb{N}_{\infty}^n$ as partial transformation of $\mathbb{N}^n$, the group of units and the subset of idempotents of the semigroup $\mathbf{I}\mathbb{N}_{\infty}^n$, the natural partial order and Green's relations on $\mathbf{I}\mathbb{N}_{\infty}^n$. In particular we show that the quotient semigroup $\m…

Entropy of Isometries Semi-groups of Hyperbolic space

We give a generalization to convex co-compact semigroups of a beautiful theorem of Patterson-Sullivan, telling that the critical exponent (that is the exponential growth rate) equals the Hausdorff dimension of the limit set (that is the smallest closed non-empty invariant subset), for a isometries discrete group of a proper hyperbolic space with compact boundary. To do that, we introduce a notion of entropy, which generalize the notion of critical exponent of discrete groups, and we show that it is equal to the upper bound of critical exponents of Schottky sub-semigroups (which are semigroups …

On the monoid of cofinite partial isometries of $\mathbb{N}$ with a bounded finite noise

In the paper we study algebraic properties of the monoid $\mathbf{I}\mathbb{N}_{\infty}^{\boldsymbol{g}[j]}$ of cofinite partial isometries of the set of positive integers $\mathbb{N}$ with the bounded finite noise $j$. For the monoids $\mathbf{I}\mathbb{N}_{\infty}^{\boldsymbol{g}[j]}$ we prove counterparts of some classical results of Eberhart and Selden describing the closure of the bicyclic semigroup in a locally compact topological inverse semigroup. In particular we show that for any positive integer $j$ every Hausdorff shift-continuous topology $τ$ on $\mathbf{I}\mathbb{N}_{\infty}^{\bo…

关于某些变换半群的支配域

On the Dominions of Certain Semigroups of Transformations

研究有限链上全变换半群中递减保序子半群的支配域，刻画了其结构并给出了元素和幂等元计数公式。

关于一个附接零的、由 $\mathbb{N}$ 的余有限部分等距构成的局部紧幺半群

On a locally compact monoid of cofinite partial isometries of $\mathbb{N}$ with adjoined zero

本文研究含 $\mathscr{C}_\mathbb{N}$ 的部分余有限等距幺半群，证明附接零后其 Hausdorff 局部紧移位连续拓扑只能为紧或离散，并推广至附接紧理想的半拓扑半群。

单调和递减有限链上的单调和递减部分变换的独异体

The monoid of monotone and decreasing partial transformations on a finite chain

本文研究了有限链上单调和递减部分变换的独异体及其相关结构，证明了其丰富性并计算了相关秩。

Groups of isometries of ultrametric Urysohn spaces and their unitary representations

We consider groups $\mathbb{I}$ of isometries of ultrametric Urysohn spaces $\mathbb{U}$. Such spaces $\mathbb{U}$ admit transparent realizations as boundaries of certain $R$-trees and the groups $\mathbb{I}$ are groups of automorphisms of these $R$-trees. Denote by $\mathbb{I}[X]\subset \mathbb{I}$ stabilizers of finite subspaces $X\subset \mathbb{U}$. Double cosets $\mathbb{I}[X]\cdot g\cdot \mathbb{I}[Y]$, where $g\in \mathbb{I}$, are enumerated by ultrametrics on union of spaces $X\cup Y$. We construct natural associative multiplications on double coset spaces $\mathbb{I}[X]\backslash \mat…

Combinatorial results for order-preserving partial injective contraction mappings

Let $ \mathcal{I}_n$ be the symmetric inverse semigroup on $X_n = \{1, 2, \ldots , n\}$. Let $\mathcal{OCI}_n$ be the subsemigroup of $\mathcal{I}_n$ consisting of all order-preserving injective partial contraction mappings, and let $\mathcal{ODCI}_n$ be the subsemigroup of $\mathcal{I}_n$ consisting of all order-preserving and order-decreasing injective partial contraction mappings of $X_n$. In this paper, we investigate the cardinalities of some equivalences on $\mathcal{OCI}_n$ and $\mathcal{ODCI}_n$ which lead naturally to obtaining the order of these semigroups. Then, we relate the formul…

关于小Schröder半群$\mathcal{SS}^{\prime}_{n}$

On the small Schröder semigroup $\mathcal{SS}^{\prime}_{n}$

本文研究了小Schröder半群的结构，证明其右丰富性并确定了其秩。

关于单射偏序Catalan幺半群

On injective partial Catalan monoids

本文研究有限链上保序降序单射部分变换构成的半群及其子半群，证明其 abundant 性质，计算 Rees 商和双边理想的秩，并刻画所有极大子半群。