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In this paper, we consider the monoids of all endomorphisms, of all weak endomorphisms, of all strong endomorphisms and of all strong weak endomorphisms of a star graph with a finite number of vertices. Our main objective is to exhibit a presentation for each of them., Comment: math.RA. arXiv admin note: text overlap with arXiv:2410.21084

In this paper, we consider the monoids of all partial endomorphisms, of all partial weak endomorphisms, of all injective partial endomorphisms, of all partial strong endomorphisms and of all partial strong weak endomorphisms of a star graph with a finite number of vertices. Our main objective is to exhibit a presentation for each of them.

In the present paper, a submonoid of the well studied monoid $POI_n$ of all order-preserving partial injections on an $n$-element chain is studied. The set $IOF_n^{par}$ of all partial transformations in $POI_n$ which are fence-preserving as well as parity-preserving form a submonoid of $POI_n$. We describe the Green's relations and ideals of $IOF_n^{par}$. For each ideal of $IOF_n^{par}$, we characterize the maximal subsemigroups. We will observe that there are three different types of maximal subsemigroups., Comment: 12 pages

In this paper we consider the monoids of all partial endomorphisms, of all partial weak endomorphisms, of all injective partial endomorphisms, of all partial strong endomorphisms and of all partial strong weak endomorphisms of a star graph with a finite number of vertices. Our main objective is to determine their ranks. We also describe their Green's relations, calculate their cardinalities and study their regularity.

A fully invarient congruence relations on the free algebra on a given type induces a variety of the given type. In contrast, a congruence relation of the free algebra provides algebra of that type. This algebra is given by a so-called presentation. In the present paper, we deal with an important class of algebras of type $(2)$, namely with semigroups of transformations on a finite set. Here, we are particularly interested in a presentation of a submonoid of the symmetric inverse monoid $I_n$. Our main result is a presentations for $IOF_n^{par}$, the monoid of all order-preserving, fence-preserving, and parity-preserving transformations on an $n$-element set., Comment: 19 pages

In this paper we consider endomorphisms of an undirected cycle graph from Semigroup Theory perspective. Our main aim is to present a process to determine sets of generators with minimal cardinality for the monoids $wEnd(C_n)$ and $End(C_n)$ of all weak endomorphisms and all endomorphisms of an undirected cycle graph $C_n$ with $n$ vertices. We also describe Green's relations and regularity of these monoids and calculate their cardinalities.

In this paper we consider the submonoids $\mathcal{OPDI}_n$, $\mathcal{MDI}_n$ and $\mathcal{ODI}_n$ of the dihedral inverse monoid $\mathcal{DI}_n$ of all orientation-preserving, monotone and order-preserving transformations, respectively. Our goal is to exhibit presentations for each of these three monoids., Comment: arXiv admin note: text overlap with arXiv:2205.02196

In this paper we consider three submonoids of the dihedral inverse monoid $\mathcal{DI}_n$, namely its submonoids $\mathcal{OPDI}_n$, $\mathcal{MDI}_n$ and $\mathcal{ODI}_n$ of all orientation-preserving, monotone and order-preserving transformations, respectively. For each of these three monoids, we compute the cardinal, give descriptions of Green's relations and determine the rank., Comment: arXiv admin note: text overlap with arXiv:2205.02196

The monoid of all partial injections on a finite set (the symmetric inverse semigroup) is of particular interest because of the well-known Wagner-Preston Theorem. In this article, we step forward the study of a submonoid of the symmetric inverse semigroup. We explore the monoid of all order-, fence-, and parity-preserving transformations on an $n$-element chain. We also characterize the transformations in that monoid and show that it has a rank $3n-6$. In particular, we provide a generating set $A_n$ of minimal size and exhibit concrete normal forms for the transformations generated by $A_n$.

For $n \in \mathbb{N}$, let $[n] = \{1, 2, \ldots, n\}$ be an $n$ - element set. As usual, we denote by $I_n$ the symmetric inverse semigroup on $[n]$, i.e. the partial one-to-one transformation semigroup on $[n]$ under composition of mappings. The crown (cycle) $\cal{C}_n$ is an $n$-ordered set with the partial order $\prec$ on $[n]$, where the only comparabilities are $$1 \prec 2 \succ 3 \prec 4 \succ \cdots \prec n \succ 1 ~~\mbox{ or }~~ 1 \succ 2 \prec 3 \succ 4 \prec \cdots \succ n \prec 1.$$ We say that a transformation $\alpha \in I_n$ is order-preserving if $x \prec y$ implies that $x\alpha \prec y\alpha$, for all $x, y $ from the domain of $\alpha$. In this paper, we study the inverse semigroup $IC_n$ of all partial automorphisms on a finite crown $\cal{C}_n$. We consider the elements, determine a generating set of minimal size and calculate the rank of $IC_n$.

For $n \in \mathbb{N}$, let $X_n = \{a_1, a_2, \ldots, a_n\}$ be an $n$ - element set and let $\textbf{F} = (X_n; <_f)$ be a fence, also called a zigzag poset. As usual, we denote by $I_n$ the symmetric inverse semigroup on $X_n$. We say that a transformation $\alpha \in I_n$ is \textit{fence-preserving} if $x <_f y$ implies that $x\alpha <_f y\alpha$, for all $x, y$ in the domain of $\alpha$. In this paper, we study the semigroup $PFI_n$ of all partial fence-preserving injections of $X_n$ and its subsemigroup $IF_n = \{\alpha \in PFI_n : \alpha^{-1}\in PFI_n\}$. Clearly, $IF_n$ is an inverse semigroup and contains all regular elements of $PFI_n.$ We characterize the Green's relations for the semigroup $IF_n$. Further, we prove that the semigroup $IF_n$ is generated by its elements with \rank$\geq n-2$. Moreover, for $n \in 2\mathbb{N}$ we find the least generating set and calculate the rank of $IF_n$.

Let $X$ be an infinite linearly ordered set and let $Y$ be a nonempty subset of $X$. We calculate the relative rank of the semigroup $OP(X,Y)$ of all orientation-preserving transformations on $X$ with restricted range $Y$ modulo the semigroup $O(X,Y)$ of all order-preserving transformations on $X$ with restricted range $Y$. For $Y = X$, we characterize the relative generating sets of minimal size.

In this paper, we study partial automorphisms and, more generally, injective partial endomorphisms of a finite undirected path from Semigroup Theory perspective. Our main objective is to give formulas for the ranks of the monoids $IEnd(P_n)$ and $PAut(P_n)$ of all injective partial endomorphisms and of all partial automorphisms of the undirected path $P_n$ with $n$ vertices. We also describe Green's relations of $PAut(P_n)$ and $IEnd(P_n)$ and calculate their cardinals.

A zig-zag order is like a directed path, only with alternating directions. A generating set of minimal size for the semigroup of all full transformations on a finite set preserving the zig-zag order was determined by Fenandes et al. in 2019. This paper deals with generating sets of the semigroup $F_{\mathbb{N}}$ of all full transformations on the set of all natural numbers preserving the zig-zag order. We prove that $F_{\mathbb{N}}$ has no minimal generating sets and present two particular infinite decreasing chains of generating sets of $F_{\mathbb{N}}$., Comment: 17 pages

In this paper, we determine the relative rank of the semigroup $T(X,Y)$ of all transformations on a finite chain $X$ with restricted range $Y \subseteq X$ modulo the set $OP(X,Y)$ of all orientation-preserving transformation in $T(X,Y)$. Moreover, we state the relative rank of the semigroup $OP(X,Y)$ modulo the set $O(X,Y)$ of all order-preserving transformations in $OP(X,Y)$. In both cases we characterize the minimal relative generating sets.

In this paper, we determine the relative rank of the semigroup OP(X) of all orientation-preserving transformations on infinite chains modulo the semigroup O(X) of all order-preserving transformations.

In this paper we study the widely considered endomorphisms and weak endomorphisms of a finite undirected path from monoid generators perspective. Our main aim is to determine the ranks of the monoids $wEnd P_n$ and $End P_n$ of all weak endomorphisms and all endomorphisms of the undirected path $P_n$ with $n$ vertices. We also consider strong and strong weak endomorphisms of $P_n$.

In this note, we consider the semigroup $O(X)$ of all order endomorphisms of an infinite chain $X$ and the subset $J$ of $O(X)$ of all transformations $\alpha$ such that $|Im(\alpha)|=|X|$. For an infinite countable chain $X$, we give a necessary and sufficient condition on $X$ for $O(X) = \langle J \rangle$ to hold. We also present a sufficient condition on $X$ for $O(X) = \langle J \rangle$ to hold, for an arbitrary infinite chain $X$.

Given an $n$-ary $k-$valued function $f$, $gap(f)$ denotes the minimal number of essential variables in $f$ which become fictive when identifying any two distinct essential variables in $f$. We particularly solve a problem concerning the explicit determination of $n$-ary $k-$valued functions $f$ with $2\leq gap(f)\leq n\leq k$. Our methods yield new combinatorial results about the number of such functions., Comment: 17 pages, Int. Conf. Algebraic and Combinatorial Coding Theory, ACCT2008, June 16 - Sunday 22, 2008, Pamporovo, BULGARIA