Idempotent Rank Research Articles

The ideals of the monoid of all orientation-preserving extensive full transformations

Let [Formula: see text] be the monoid of all orientation-preserving and extensive full transformations on [Formula: see text]. In this paper, we compute the rank and the idempotent rank of the ideals of the monoid [Formula: see text]. Moreover, we determine the maximal subsemigroups as well as the maximal subsemibands of the ideals of the monoid [Formula: see text]. Our work extends previous results found in the literature.

Idempotent Rank of Identity Difference Transformation Semigroup

The identity difference transformation semigroup was examined for idempotent rank and the combinatorial results for idempotent ranks of IDT_n,IDO_n,IDI_n,and IDPOI_n were obtained.

The monoid of all orientation-preserving and extensive full transformations on a finite chain

Let [Formula: see text] be the monoid of all orientation-preserving and extensive full transformations on [Formula: see text] ordered in the standard way. In this paper, we determine the minimum generating set and the minimum idempotent generating set of [Formula: see text], and so the rank and the idempotent rank of [Formula: see text] are obtained. Moreover, we describe maximal subsemigroups and maximal idempotent generated subsemigroups of [Formula: see text] and completely obtain their classifications.

Sandwich semigroups in locally small categories II: transformations

Fix sets X and Y, and write $$\mathcal P\mathcal T_{XY}$$ for the set of all partial functions $$X\rightarrow Y$$ . Fix a partial function $${a:Y\rightarrow X}$$ , and define the operation $$\star _a$$ on $$\mathcal P\mathcal T_{XY}$$ by $$f\star _ag=fag$$ for $$f,g\in \mathcal P\mathcal T_{XY}$$ . The sandwich semigroup $$(\mathcal P\mathcal T_{XY},\star _a)$$ is denoted $$\mathcal P\mathcal T_{XY}^a$$ . We apply general results from Part I to thoroughly describe the structural and combinatorial properties of $$\mathcal P\mathcal T_{XY}^a$$ , as well as its regular and idempotent-generated subsemigroups, $${\text {Reg}}(\mathcal P\mathcal T_{XY}^a)$$ and $$\mathbb E(\mathcal P\mathcal T_{XY}^a)$$ . After describing regularity, stability and Green’s relations and preorders, we exhibit $${\text {Reg}}(\mathcal P\mathcal T_{XY}^a)$$ as a pullback product of certain regular subsemigroups of the (non-sandwich) partial transformation semigroups $$\mathcal P\mathcal T_X$$ and $$\mathcal P\mathcal T_Y$$ , and as a kind of “inflation” of $$\mathcal P\mathcal T_A$$ , where A is the image of the sandwich element a. We also calculate the rank (minimal size of a generating set) and, where appropriate, the idempotent rank (minimal size of an idempotent generating set) of $$\mathcal P\mathcal T_{XY}^a$$ , $${\text {Reg}}(\mathcal P\mathcal T_{XY}^a)$$ and $$\mathbb E(\mathcal P\mathcal T_{XY}^a)$$ . The same program is also carried out for sandwich semigroups of totally defined functions and for injective partial functions. Several corollaries are obtained for various (non-sandwich) semigroups of (partial) transformations with restricted image, domain and/or kernel.

Twisted Brauer monoids

We investigate the structure of the twisted Brauer monoid , comparing and contrasting it with the structure of the (untwisted) Brauer monoid . We characterize Green's relations and pre-orders on , describe the lattice of ideals and give necessary and sufficient conditions for an ideal to be idempotent generated. We obtain formulae for the rank (smallest size of a generating set) and (where applicable) the idempotent rank (smallest size of an idempotent generating set) of each principal ideal; in particular, when an ideal is idempotent generated, its rank and idempotent rank are equal. As an application of our results, we describe the idempotent generated subsemigroup of (which is not an ideal), as well as the singular ideal of (which is neither principal nor idempotent generated), and we deduce that the singular part of the Brauer monoid is idempotent generated, a result previously proved by Maltcev and Mazorchuk.

On the ranks of the semigroup of A-decreasing finite transformations

For [Formula: see text], let [Formula: see text] be the semigroup of all singular mappings on [Formula: see text]. For each nonempty subset [Formula: see text] of [Formula: see text], let [Formula: see text] be the semigroup of all [Formula: see text]-decreasing mappings on [Formula: see text]. In this paper we determine the rank and idempotent rank of the semigroup [Formula: see text].

On the semigroups of order-preserving transformations generated by idempotents of rank n −1

Let [Formula: see text] be the semigroup of all singular order-preserving mappings on the finite set [Formula: see text]. It is known that [Formula: see text] is generated by its set of idempotents of rank [Formula: see text], and its rank and idempotent rank are [Formula: see text] and [Formula: see text], respectively. In this paper, we study the structure of the semigroup generated by any nonempty subset of idempotents of rank [Formula: see text] in [Formula: see text]. We also calculate its rank and idempotent rank.

Diagram monoids and Graham–Houghton graphs: Idempotents and generating sets of ideals

We study the ideals of the partition, Brauer, and Jones monoid, establishing various combinatorial results on generating sets and idempotent generating sets via an analysis of their Graham–Houghton graphs. We show that each proper ideal of the partition monoid Pn is an idempotent generated semigroup, and obtain a formula for the minimal number of elements (and the minimal number of idempotent elements) needed to generate these semigroups. In particular, we show that these two numbers, which are called the rank and idempotent rank (respectively) of the semigroup, are equal to each other, and we characterize the generating sets of this minimal cardinality. We also characterize and enumerate the minimal idempotent generating sets for the largest proper ideal of Pn, which coincides with the singular part of Pn. Analogous results are proved for the ideals of the Brauer and Jones monoids; in each case, the rank and idempotent rank turn out to be equal, and all the minimal generating sets are described. We also show how the rank and idempotent rank results obtained, when applied to the corresponding twisted semigroup algebras (the partition, Brauer, and Temperley–Lieb algebras), allow one to recover formulae for the dimensions of their cell modules (viewed as cellular algebras) which, in the semisimple case, are formulae for the dimensions of the irreducible representations of the algebras. As well as being of algebraic interest, our results relate to several well-studied topics in graph theory including the problem of counting perfect matchings (which relates to the problem of computing permanents of {0,1}-matrices and the theory of Pfaffian orientations), and the problem of finding factorizations of Johnson graphs. Our results also bring together several well-known number sequences such as Stirling, Bell, Catalan and Fibonacci numbers.

Motzkin monoids and partial Brauer monoids

We study the partial Brauer monoid and its planar submonoid, the Motzkin monoid. We conduct a thorough investigation of the structure of both monoids, providing information on normal forms, Green's relations, regularity, ideals, idempotent generation, minimal (idempotent) generating sets, and so on. We obtain necessary and sufficient conditions under which the ideals of these monoids are idempotent-generated. We find formulae for the rank (smallest size of a generating set) of each ideal, and for the idempotent rank (smallest size of an idempotent generating set) of the idempotent-generated subsemigroup of each ideal; in particular, when an ideal is idempotent-generated, the rank and idempotent rank are equal. Along the way, we obtain a number of results of independent interest, and we demonstrate the utility of the semigroup theoretic approach by applying our results to obtain new proofs of some important representation theoretic results concerning the corresponding diagram algebras, the partial (or rook) Brauer algebra and Motzkin algebra.

The (p,q)-potent ranks of certain semigroups of transformations

Let [Formula: see text] and [Formula: see text] be the partial transformation and the strictly partial transformation semigroups on the finite set [Formula: see text]. It is well known that the ranks of the semigroups [Formula: see text] and [Formula: see text] are [Formula: see text], for [Formula: see text], and [Formula: see text], for [Formula: see text], respectively. The idempotent rank, defined as the smallest number of idempotents generating set, of the semigroup [Formula: see text] has the same value as the rank. Idempotent can be seen as a special case (with [Formula: see text]) of [Formula: see text]-potent. In this paper, we determine the [Formula: see text]-potent ranks, defined as the smallest number of [Formula: see text]-potents generating set, of the semigroups [Formula: see text], for [Formula: see text], and [Formula: see text], for [Formula: see text].

Idempotent Generation in the Endomorphism Monoid of a Uniform Partition

Denote by 𝒯n and 𝒮n the full transformation semigroup and the symmetric group on the set {1,…, n}, and ℰn = {1} ∪ (𝒯n∖𝒮n). Let 𝒯(X, 𝒫) denote the monoid of all transformations of the finite set X preserving a uniform partition 𝒫 of X into m subsets of size n, where m, n ≥ 2. We enumerate the idempotents of 𝒯(X, 𝒫), and describe the submonoid S = ⟨ E ⟩ generated by the idempotents E = E(𝒯(X, 𝒫)). We show that S = S1 ∪ S2, where S1 is a direct product of m copies of ℰn, and S2 is a wreath product of 𝒯n with 𝒯m∖𝒮m. We calculate the rank and idempotent rank of S, showing that these are equal, and we also classify and enumerate all the idempotent generating sets of minimal size. In doing so, we also obtain new results about arbitrary idempotent generating sets of ℰn.

Variants of finite full transformation semigroups

The variant of a semigroup [Formula: see text] with respect to an element [Formula: see text], denoted [Formula: see text], is the semigroup with underlying set [Formula: see text] and operation ⋆ defined by [Formula: see text] for [Formula: see text]. In this paper, we study variants [Formula: see text] of the full transformation semigroup [Formula: see text] on a finite set [Formula: see text]. We explore the structure of [Formula: see text] as well as its subsemigroups [Formula: see text] (consisting of all regular elements) and [Formula: see text] (consisting of all products of idempotents), and the ideals of [Formula: see text]. Among other results, we calculate the rank and idempotent rank (if applicable) of each semigroup, and (where possible) the number of (idempotent) generating sets of the minimal possible size.

IDEMPOTENT RANK IN THE ENDOMORPHISM MONOID OF A NONUNIFORM PARTITION

We calculate the rank and idempotent rank of the semigroup ${\mathcal{E}}(X,{\mathcal{P}})$ generated by the idempotents of the semigroup ${\mathcal{T}}(X,{\mathcal{P}})$ which consists of all transformations of the finite set $X$ preserving a nonuniform partition ${\mathcal{P}}$. We also classify and enumerate the idempotent generating sets of minimal possible size. This extends results of the first two authors in the uniform case.

ON THE m-POTENT RANKS OF CERTAIN SEMIGROUPS OF ORIENTATION PRESERVING TRANSFORMATIONS

It is known that the ranks of the semigroups <TEX>$\mathcal{SOP}_n$</TEX>, <TEX>$\mathcal{SPOP}_n$</TEX> and <TEX>$\mathcal{SSPOP}_n$</TEX> (the semigroups of orientation preserving singular self-maps, partial and strictly partial transformations on <TEX>$X_n={1,2,{\ldots},n}$</TEX>, respectively) are n, 2n and n + 1, respectively. The idempotent rank, defined as the smallest number of idempotent generating set, of <TEX>$\mathcal{SOP}_n$</TEX> and <TEX>$\mathcal{SSPOP}_n$</TEX> are the same value as the rank, respectively. Idempotent can be seen as a special case (with m = 1) of m-potent. In this paper, we investigate the m-potent ranks, defined as the smallest number of m-potent generating set, of the semigroups <TEX>$\mathcal{SOP}_n$</TEX>, <TEX>$\mathcal{SPOP}_n$</TEX> and <TEX>$\mathcal{SSPOP}_n$</TEX>. Firstly, we characterize the structure of the minimal generating sets of <TEX>$\mathcal{SOP}_n$</TEX>. As applications, we obtain that the number of distinct minimal generating sets is <TEX>$(n-1)^nn!$</TEX>. Secondly, we show that, for <TEX>$1{\leq}m{\leq}n-1$</TEX>, the m-potent ranks of the semigroups <TEX>$\mathcal{SOP}_n$</TEX> and <TEX>$\mathcal{SPOP}_n$</TEX> are also n and 2n, respectively. Finally, we find that the 2-potent rank of <TEX>$\mathcal{SSPOP}_n$</TEX> is n + 1.

On the Semigroups of Order-preserving and A-Decreasing Finite Transformations

For n ∈ ℕ, let On be the semigroup of all singular order-preserving mappings on [n]= {1,2,…,n}. For each nonempty subset A of [n], let On(A) = {α ∈ On: (∀ k ∈A) kα ≤ k} be the semigroup of all order-preserving and A-decreasing mappings on [n]. In this paper it is shown that On(A) is an abundant semigroup with n-1𝒟*-classes. Moreover, On(A) is idempotent-generated and its idempotent rank is 2n-2- |A\ {n}|. Further, it is shown that the rank of On(A) is equal to n-1 if 1 ∈ A, and it is equal to n otherwise.

Rank and idempotent rank of finite full transformation semigroups with restricted range

Let T(X) be the full transformation semigroup on the set X and let T(X,Y) be the semigroup consisting of all total transformations from X into a fixed nonempty subset Y of X. In 2011, Sanwong studied the regular part $$F(X,Y)=\bigl\{\alpha\in T(X,Y): X\alpha\subseteq Y\alpha\bigr\}, $$ of T(X,Y) and described its Green’s relations and ideals. In this paper, we compute the rank of F(X,Y) when X is a finite set. Moreover, we obtain the rank and idempotent rank of its ideals.

On the Ranks of Certain Semigroups of Orientation Preserving Transformations

In this article, we investigate the ranks of SOP n , SPOP n , and SSPOP n (the semigroups of orientation preserving singular selfmaps, partial, and strictly partial transformations on [n] = {1, 2,…, n}, respectively). Firstly, we show that the rank and idempotent rank of SOP n are n. Secondly, we characterize the structure of the idempotent-generating sets of SPOP n , and prove that the rank and idempotent rank of SPOP n are 2n. Finally, we find that the rank of SSPOP n is n + 1. This research extends the results of Gomes and Howie [4] on the ranks of the semigroups O n , PO n , and SPO n (the semigroups of order preserving singular selfmaps, partial, and strictly partial transformations on [n] = {1, 2,…, n}, respectively).

Rank Properties of the Semigroup of Singular Transformations on a Finite Set

It is known that the semigroup Sing n of all singular self-maps of X n = {1,2,…, n} has rank n(n − 1)/2. The idempotent rank, defined as the smallest number of idempotents generating Sing n , has the same value as the rank. (See Gomes and Howie, 1987.) Idempotents generating Sing n can be seen as special cases (with m = r = 2) of (m, r)-path-cycles, as defined in Ay\\i k et al. (2005). The object of this article is to show that, for fixed m and r, the (m, r)-rank of Sing n , defined as the smallest number of (m, r)-path-cycles generating Sing n , is once again n(n − 1)/2.

Idempotent rank in endomorphism monoids of finite independence algebras

In 1992, Fountain and Lewin showed that any proper ideal of an endomorphism monoid of a finite independence algebra is generated by idempotents. Here the ranks and idempotent ranks of these ideals are determined. In particular, it is shown that when the algebra has dimension greater than or equal to three the idempotent rank equals the rank.

On Idempotent Ranks of Semigroups of Partial Transformations

A subset U of a semigroup S is a generating set for S if every element of S may be written as a finite product of elements of U. The rank of S is the size of a minimal generating set of S, and the idempotent rank of S is the size of a minimal generating set of S consisting of idempotents in S. A partition of a q-element subset of the set Xn={1,2,..., n} is said to be of type τ if the sizes of its classes form the partition τ of q ≤ n. A non-trivial partition τ of a positive integer q consists of k < q elements. For a non-trivial partition τ of q ≤ n, the semigroup S(τ), generated by all the transformations with kernels of type τ, is idempotent-generated. It is known that if τ is a non-trivial partition of n, that is, S(τ) consists of total many-to-one transformations, then the rank and the idempotent rank of S(τ) are both equal to max{nd, N(τ)}, where N(τ) is the number of partitions of Xn of type τ. We extend this result to semigroups of partial transformations, and prove that if τ is a non-trivial partition of q < n, then the rank and the idempotent rank of S(τ) are both equal to N(τ).