Let [Formula: see text] be the symmetric inverse semigroup on the finite chain [Formula: see text] and let [Formula: see text] for [Formula: see text]. A quasi-idempotent element is an element [Formula: see text] with the property that [Formula: see text]. In this paper, we obtain a useful method by listing the subsets of [Formula: see text] to build a (minimal) quasi-idempotent generating set of [Formula: see text] both as a semigroup and also as an inverse semigroup for [Formula: see text] and [Formula: see text].

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 formulae obtained to Fibonacci numbers. Similar results about $\mathcal{ORCI}_n$, the semigroup of order-preserving or order-reversing injective partial contraction mappings, are deduced.

This work analyses the algebraic properties of the sub-semi group of partial Isometries (〖DP〗_n) and of order preserving partial isometries (〖ODP〗_n) of a finite chain X_n={1,2,…n}, with a symmetric inverse semigroup I_n defined on it. It also investigates the subsemigroups 〖DP〗_n and 〖ODP〗_n for cycle structure and shows that 〖ODP〗_n is a 0 -E- unitary inverse semigroup.

In the present paper, we study a submonoid of the symmetric inverse semigroup \(I_n\). Specifically, we consider the monoid of all order-, fence-, and parity-preserving transformations of \(I_n\). While the rank and a set of generators of minimal size for this monoid are already known, we will provide a presentation for this monoid.

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. Let [Formula: see text] be a positive natural number and [Formula: see text] be the semigroup of all fence-preserving partial one-to-one maps of [Formula: see text] into itself with respect to composition of maps and the fence [Formula: see text]. There is considered the inverse semigroup [Formula: see text] of all [Formula: see text] such that [Formula: see text] is regular in [Formula: see text], order-preserving with respect to the order [Formula: see text] and parity-preserving. According to the main result of the paper, it is [Formula: see text] the least of the cardinalities of the generating sets of [Formula: see text] for [Formula: see text]. There is determined a concrete representation of a generating set of minimal size.

A semigroup S is called left ample if it can be embedded in the symmetric inverse semigroup IX of partial bijections of a non-empty set X such that the image of S is closed under the unary operation α → αα⁻¹, where α⁻¹ is the inverse of α in IX. Right ample semigroups are defined dually. A semigroup is called ample if it is both left and right ample. A monoid is (left, right) ample if it is (left, right) ample as a semigroup. We observe that the dominion of an ample subsemigroup of IX coincides with the inverse subsemigroup of IX generated by it. We then determine the dominions of certain submonoids of In, the symmetric inverse semigroup over a finite chain 1<2<⋯<n.

Publisher Correction to: Topologies on the symmetric inverse semigroup

The symmetric inverse semigroup I(X) on a set X is the collection of all partial bijections between subsets of X with composition as the algebraic operation. We study the minimal Hausdorff inverse semigroup topology on I(X). We present some characterizations of it. When X is countable such topology is Polish.

Supposed is a finite set, then a function is called a finite partial transformation semigroup , which moves elements of from its domain to its co-domain by a distance of where . The total work done by the function is therefore the sum of these distances. It is a known fact that and . In this this research paper, we have mainly presented the numerical solutions of the total work done, the average work done by functions on the finite symmetric inverse semigroup of degree , and the finite full transformation semigroup of degree , as well as their respective powers for a given fixed time in space. We used an effective methodology and valid combinatorial results to generalize the total work done, the average work done and powers of each of the transformation semigroups. The generalized results were tested by substituting small values of and a specified fixed times in space. Graphs were plotted in each case to illustrate the nature of the total work done and the average work done. The results obtained in this research article have an important application in some branch of physics and theoretical computer science

Quasi-idempotent ranks of the proper ideals in finite symmetric inverse semigroups

A note on feebly compact semitopological symmetric inverse semigroups of a bounded finite rank

For an integer let be the symmetric inverse semigroup of partial injective transformations on an n-element set. For denote by and respectively, the first and second centralizer of λ in We determine the structure of and in terms of Green’s relations, including the partial order of -classes, and express as a direct product of cyclic groups with zero adjoined and a monogenic monoid. For each individual Green relation we determine such that in is inherited from and λ such that all Green relations in are inherited from We also provide a representation of the partial order of -classes in as a known lattice, and describe such that which gives a class of maximal commutative subsemigroups of

A complete description of indecomposable characters of the infinite symmetric inverse semigroup is given. The method essentially uses the decomposition of the elements of this semigroup into a product of independent quasi-cycles and the multiplicativity theorem. Realizations of all factor representations of finite type are constructed.

Let I_n, S_n and A_n be the symmetric inverse semigroup, the symmetric group and the alternating group on X_n={1,…,n}, for n≥2, respectively. Also let I_(n,r) be the subsemigroup consists of all partial injective maps with height less than or equal to r for 1≤r≤n-1, and let SI_(n,r)=I_(n,r)∪S_n and AI_(n,r)=I_(n,r)∪A_n. A non-idempotent element whose square is an idempotent is called a quasi-idempotent. In this paper we obtain the rank and the quasi-idempotent rank of SI_(n,r) (of AI_(n,r)). Also we obtain the relative rank and the relative quasi-idempotent rank of SI_(n,r) modulo S_n (of AI_(n,r) modulo A_n).

We study the semigroup extension $\mathscr{I}_\lambda^n(S)$ of a semigroup $S$ by symmetric inverse semigroups of a bounded finite rank. We describe idempotents and regular elements of the semigroups $\mathscr{I}_\lambda^n(S)$ and $\overline{\mathscr{I}_\lambda^n}(S)$ show that the semigroup $\mathscr{I}_\lambda^n(S)$ ($\overline{\mathscr{I}_\lambda^n}(S)$) is regular, orthodox, inverse or stable if and only if so is $S$. Green's relations are described on the semigroup $\mathscr{I}_\lambda^n(S)$ for an arbitrary monoid $S$. We introduce the conception of a semigroup with strongly tight ideal series, and proved that for any infinite cardinal $\lambda$ and any positive integer $n$ the semigroup $\mathscr{I}_\lambda^n(S)$ has a strongly tight ideal series provides so has $S$. At the finish we show that for every compact Hausdorff semitopological monoid $(S,\tau_S)$ there exists a unique its compact topological extension $\left(\mathscr{I}_\lambda^n(S),\tau_{\mathscr{I}}^\mathbf{c}\right)$ in the class of Haudorff semitopological semigroups.

Quiver mutations and Boolean reflection monoids

Conjugacy in inverse semigroups

We construct an extended quantum spin chain model by introducing new degrees of freedom to the Fredkin spin chain. The new degrees of freedom called arrow indices are partly associated to the symmetric inverse semigroup $\cS^3_1$. Ground states of the model fall into three different phases, and quantum phase transition takes place at each phase boundary. One of the phases exhibits logarithmic violation of the area law of entanglement entropy and quantum criticality, whereas the other two obey the area law. As an interesting feature arising by the extension, there are excited states due to disconnections with respect to the arrow indices. We show that these states are localized without disorder.

Area law violations for entanglement entropy in the form of a square root has recently been studied for one-dimensional frustration-free quantum systems based on the Motzkin walks and their variations. Here, we further modify the Motzkin walks using the elements of a symmetric inverse semigroup as basis states on each step of the walk. This change alters the number of paths allowed in the Motzkin walks and by introducing an appropriate term in the Hamiltonian with a tunable parameter we show that we can jump from a state that violates the area law logarithmically to a state that obeys the area law providing an example of quantum phase transition in a one-dimensional system.

Let $$I_{n}$$ be the symmetric inverse semigroup on $$X_{n}=\{1,\ldots ,n\}$$ , and let $$DP_{n}$$ and $$ODP_{n}$$ be its subsemigroups of partial isometries and of order-preserving partial isometries on $$X_{n}$$ under its natural order, respectively. In this paper we find the ranks of the subsemigroups $$DP_{n,r}=\{ \alpha \in DP_{n}:|\mathrm {im\, }(\alpha )|\le r\}$$ and $$ODP_{n,r}=\{ \alpha \in ODP_{n}: |\mathrm {im\, }(\alpha )|\le r\}$$ for $$2\le r\le n-1$$ .