Related papers: Presentations for three remarkable submonoids of t…
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…
In this paper we give presentations for the monoid $\mathcal{DP}_n$ of all partial isometries on $\{1,\ldots,n\}$ and for its submonoid $\mathcal{ODP}_n$ of all order-preserving partial isometries.
The dual symmetric inverse monoid $\mathscr{I}_n^*$ is the inverse monoid of all isomorphisms between quotients of an $n$-set. We give a monoid presentation of $\mathscr{I}_n^*$ and, along the way, establish criteria for a monoid to be…
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…
In this paper, we consider the inverse submonoids $AM_n$ of monotone transformations and $AO_n$ of order-preserving transformations of the alternating inverse monoid $AI_n$ on a chain with $n$ elements. We compute the cardinalities,…
We study permutations on n elements preserving orientation (parity) of every subset of size k. We describe all groups of these permutations. Unexpectedly, these groups (except for some special cases) are either trivial, cyclic or dihedral.…
In this paper, we consider the inverse submonoids $AOR_n$ of oriented transformations and $AOP_n$ of orientation-preserving transformations of the alternating inverse monoid $AI_n$ on a chain with $n$ elements. We compute the cardinalities,…
In this note we give a presentation for the monoid $IO_n$ of all order-preserving transformations of a $n$-chain whose ranges are intervals. We also consider the submonoid $IO_n^-$ of $IO_n$ consisting of order-decreasing transformations,…
In this paper we study the cyclic inverse monoid $\CI_n$ on a set $\Omega_n$ with $n$ elements, i.e. the inverse submonoid of the symmetric inverse monoid on $\Omega_n$ consisting of all restrictions of the elements of a cyclic subgroup of…
A recent paper studied an inverse submonoid $M_n$ of the rook monoid, by representing the nonzero elements of $M_n$ via certain triplets belonging to $\mathbb{Z}^3$. In this short note, we allow the triplets to belong to $\mathbb{R}^3$. We…
In this paper we compute the rank and exhibit a presentation for the monoids of all $P$-stable and $P$-order preserving partial permutations on a finite set $\Omega$, with $P$ an ordered uniform partition of $\Omega$. These (inverse)…
In this note we consider the monoid $\mathcal{PODI}_n$ of all monotone partial permutations on $\{1,\ldots,n\}$ and its submonoids $\mathcal{DP}_n$, $\mathcal{POI}_n$ and $\mathcal{ODP}_n$ of all partial isometries, of all order-preserving…
We give a monoid presentation in terms of generators and defining relations for the partial analogue of the finite dual inverse symmetric monoid.
In this paper, we characterize the monoid of endomorphisms of the semigroup of all oriented full transformations of a finite chain, as well as the monoid of endomorphisms of the semigroup of all oriented partial transformations and the…
By the theorems of Cayley and Vagner-Preston, the full transformation monoids and the symmetric inverse monoids play analogous roles in the theory of monoids and inverse monoids, as the symmetric groups do in the theory of groups. Every…
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 $[n]$,…
In this thesis we study the subsemigroup structure of the symmetric inverse monoid $I_X$, the inverse semigroup of bijections between subsets of the set $X$, when $X$ is an infinite set. We explore three different approaches to this task.…
We introduce the inverse monoid of inner partial automorphisms of a semigroup -- a tool that associates to every semigroup an inverse semigroup. When the semigroup is a group, this inverse semigroup is isomorphic to the group of inner…
Wreath products involving symmetric inverse monoids/semigroups/categories arise in many areas of algebra and science, and presentations by generators and relations are crucial tools in such studies. The current paper finds such…
In this short note, we show that the number of monogenic submonoids of the full transformation monoid of degree $n$ for $n > 0$, equals the sum of the number of cyclic subgroups of the symmetric groups on $1$ to $n$ points. We also prove an…