Narrow Results

Let $X$ be a nonempty set and let ${𝒫}{𝒯}(X)$ be the monoid of all partial transformations of $X$. Given a nonempty subset $Y$ of $X$, we denote by ${𝒫}{𝒯}(X,Y)$ the subsemigroup of ${𝒫}{𝒯}(X)$ of all partial transformations with range contained in $Y$. In 2014, Fernandes and Sanwong [On the ranks of semigroups of transformations on a finite set with restricted range, Algebra Colloquium. 21(3) (2014) 497–510] proved that the semigroup ${𝒫}{ℱ}(X,Y)=\{\alpha \in {𝒫}{𝒯}(X,Y):X\alpha =Y\alpha \}$ is the largest regular subsemigroup of ${𝒫}{𝒯}(X,Y)$ and showed that this subsemigroup determines Green’s relations on ${𝒫}{𝒯}(X,Y)$. In this paper, we classify completely the maximal subsemigroups of ${𝒫}{ℱ}(X,Y)$ when $\left|Y\right|=r$. Moreover, we prove that the rank of ${𝒫}{ℱ}(X,Y)$ is equal to ${(r+1)}^{n-r}+2$ when $\left|X\right|=n$ and $\left|Y\right|=r$.

Let $k$ be a field of arbitrary characteristic, $A$ be a domain, and K = frac(A). Then

• ⁽¹⁾All exponential maps of $k^{[3]}$ are rigid, and we give a necessary and sufficient condition for the triangularity of $\delta \in \mathrm{\text{EXP}}(k^{[3]})$.
• ⁽²⁾If $\delta \in \mathrm{\text{EXP}}(A^{[3]})$ such that $\mathrm{\text{rank}}(\delta )=\mathrm{\text{rank}}({\delta }_K)$, then $\delta$ is rigid and we give a necessary and sufficient condition for the triangularity of $\delta$.

When $k$ is of zero characteristic, $(1)$ is due to [D. Daigle, A necessary and sufficient condition for triangulability of derivations of $k[X,Y,Z]$, J. Pure Appl. Algebra113(3) (1996) 297–305] and $(2)$ is due to [M. K. Keshari and S. A. Lokhande, A note on rigidity and triangulability of a derivation, J. Commut. Algebra6(1) (2014) 95–100].

Let $1\le k\le n$. We denote by ${{𝒪}{ℰ}}_{n,k}$ the monoid of all order-preserving k-extensive full transformations on $\{1,\dots ,n\}$ ordered in the standard way. In this paper, it is shown that ${{𝒪}{ℰ}}_{n,k}$ is not a regular semigroup in general, but it is an abundant semigroup. Moreover, we prove that ${{𝒪}{ℰ}}_{n,k}$ is idempotent-generated and compute the idempotent rank and rank of the monoid ${{𝒪}{ℰ}}_{n,k}$. Moreover, we determine the maximal idempotent generated subsemigroups of the monoid ${{𝒪}{ℰ}}_{n,k}$.

Given a partially commutative alphabet and a set of words $L$, the rank of $L$ expresses the amount of shuffling required to produce a word belonging to $L$ from two words whose concatenation belongs to the closure of $L$ with respect to the partial commutation. In this paper, the notion of rank is generalized from concatenations of two words to an arbitrary fixed number of words. In this way, an infinite sequence of non-negative integers and infinity is assigned to every set of words. It is proved that in the case of alphabets defining free commutative monoids, as well as in the more general case of direct products of free monoids, sequences of ranks of regular sets are exactly non-decreasing sequences that are eventually constant. On the other hand, by uncovering a relationship between rank sequences of regular sets and rational series over the min-plus semiring, it is shown that already for alphabets defining free products of free commutative monoids, rank sequences need not be eventually periodic.

We simulate sytems up to 10 001 × 10 001 × 10 001 in three dimensions at the percolation threshold p_c of 0.31160. We find that the fractal dimension is ≃2.53±0.02, the cluster size distribution exponent, τ, is 2.186±0.002 and an exponent of 0.85 describing how the mass of the clusters scale with rank. Corrections-to-scaling exponents of ≃-0.7 are observed for n_s and for the mass of the largest cluster. We also check the percolation threshold and report good agreement with recent values.

In this paper, a general architecture for analog implementation of loser/winner-take-all (LTA/WTA) and other rank order circuits is presented. This architecture is composed of a differential amplifier with merged n-inputs and a merged common-source with active load (MCSAL) circuit to choose the desired input. The advantages of the proposed structure are simplicity, very high resolution, very low supply voltage requirements, very low output resistor, low power dissipation, low active area and simple expansion for multiple inputs by adding only three transistors for each extra input. The post-layout simulation results of proposed circuits are presented by HSPICE software in 0.35-μm CMOS process technology. The total power dissipation of proposed circuits is about 110-μW. Also, the total active area is about 550-μm² for five-input proposed circuits, and would be negligibly increased for each extra input.

Let h₁, h₂,… be a sequence of elements in a free group and let H be the subgroup they generate. Let H′ be the subgroup generated by w₁, w₂, …, where each w_i is a word in h_i and possibly other h_j, such that the associated directed graph has the finite paths property. We show that rank H′≥ rank H. As a corollary we get that , where is the subgroup generated by the roots of the elements in H. If H₀ is finitely generated and the sequence of subgroups H₀, H₁, H₂, … satisfies then the sequence stabilizes, i.e. for some m, H_i=H_i+1 for every i≥ m. When applied to systems of equations in free groups, we give conditions on a transformation of the system such that the maximal rank of a solution (the inner rank) does not increase. In particular, we show that if in "Lyndon equation" the exponents a_i satisfy gcd(a₁,…,a_n)≠1 then the inner rank is ⌊ n/2⌋. The proofs are mostly elementary.

We answer a question due to A. Myasnikov by proving that all expected ranks occur as the ranks of intersections of finitely generated subgroups of free groups.

The variant of a semigroup $S$ with respect to an element $a\in S$, denoted $S^a$, is the semigroup with underlying set $S$ and operation ⋆ defined by $x\star y=xay$ for $x,y\in S$. In this paper, we study variants ${{𝒯}}_X^a$ of the full transformation semigroup ${{𝒯}}_X$ on a finite set $X$. We explore the structure of ${{𝒯}}_X^a$ as well as its subsemigroups $\mathrm{\text{Reg}}({{𝒯}}_X^a)$ (consisting of all regular elements) and ${{ℰ}}_X^a$ (consisting of all products of idempotents), and the ideals of $\mathrm{\text{Reg}}({{𝒯}}_X^a)$. 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.

In this note, we generalize a theorem of Juan Souto on rank and Nielsen equivalence in the fundamental group of a hyperbolic fibered $3$-manifold to a large class of hyperbolic group extensions. This includes all hyperbolic extensions of surfaces groups as well as hyperbolic extensions of free groups by convex cocompact subgroups of $\mathrm{\text{Out}}(F_n)$.

We give a thorough structural analysis of the principal one-sided ideals of arbitrary semigroups, and then apply this to full transformation semigroups and symmetric inverse monoids. One-sided ideals of these semigroups naturally occur as semigroups of transformations with restricted range or kernel.

We determine a large class of links, including some satellite and hyperbolic links, for each of which the tunnel number is the minimum possible, the number of its components minus one, and observe that the rank versus genus conjecture is valid for this class of links.

Let κ be a singular cardinal. Karp's notion of a chain model of size κ is defined to be an ordinary model of size κ along with a decomposition of it into an increasing union of length cf(κ). With a notion of satisfaction and (chain)-isomorphism such models give an infinitary logic largely mimicking first order logic. In this paper we associate to this logic a notion of a dynamic EF-game which gauges when two chain models are chain-isomorphic. To this game is associated a tree which is a tree of size κ with no κ-branches (even no cf(κ)-branches). The measure of how non-isomorphic the models are is reflected by a certain order on these trees, called reduction. We study the collection of trees of size κ with no κ-branches under this notion and prove that when cf(κ) = ω this collection is rather regular; in particular it has universality number exactly κ⁺. Such trees are then used to develop a descriptive set theory of the space ^cf(κ)κ.

The main result of the paper gives in the case of κ strong limit singular an exact connection between the descriptive set-theoretic complexity of the chain isomorphism orbit of a model, the reduction order on the trees and winning strategies in the corresponding dynamic EF games. In particular we obtain a neat analog of the notion of Scott watershed from the Scott analysis of countable models.

Let G be a finite group and X a conjugacy class of G. We denote rank(G:X) to be the minimum number of elements of X generating G. In the present article, we determine the ranks of the Conway groups Co₂ and Co₃. Computations were carried with the aid of computer algebra system 𝔾𝔸ℙ [18].

An R-module _RM is called morphic if M/im α ≅ ker α for every endomorphism α of M, that is, if the dual of the Noether isomorphism theorem holds. Mostly all morphic Z-modules are determined leaving open some classes of nonsplitting mixed groups, which actually cannot be completely characterized. However, for these classes, comprehensive information including negative examples is given.

It is known that the ranks of the semigroups 𝒮ing_n and 𝒮𝒫𝒯_n (the semigroups of singular self-maps and partial transformations on X_n = {1, 2, …, n}, respectively) are , respectively. The idempotent ranks, defined as the smallest number of idempotents generating set, of 𝒮ing_n and 𝒮𝒫𝒯_n are the same value as the ranks, respectively. Idempotent can be seen as a special case (with m = r = 1) of (m, r)-potent. In this paper, we determine the (m, r)-potent ranks, defined as the smallest number of (m, r)-potent generating set, of the semigroups 𝒮ing_n and 𝒮𝒫𝒯_n.

Let ${{𝒪}}_n$ be the semigroup of all singular order-preserving mappings on the finite set $X_n=\{1,2,\dots ,n\}$. It is known that ${{𝒪}}_n$ is generated by its set of idempotents of rank $n-1$, and its rank and idempotent rank are $n$ and $2n-2$, respectively. In this paper, we study the structure of the semigroup generated by any nonempty subset of idempotents of rank $n-1$ in ${{𝒪}}_n$. We also calculate its rank and idempotent rank.

In this paper, we survey the main known constructions of Ferrers diagram rank-metric codes, and establish new results on a related conjecture by Etzion and Silberstein. We also give a sharp lower bound on the dimension of linear rank-metric anticodes with a given profile. Combining our results with the multilevel construction, we produce examples of subspace codes with the largest known cardinality for the given parameters. We also apply results from algebraic geometry to the study of the analogous problem over an algebraically closed field, proving that the bound by Etzion and Silberstein can be improved in this case, and providing a sharp bound for full-rank matrices.

Let ${{𝒫}}_n$ and ${{𝒮}{𝒫}}_n$ be the partial transformation and the strictly partial transformation semigroups on the finite set $X_n=\{1,2,\dots ,n\}$. It is well known that the ranks of the semigroups ${𝒫}(n,r)=\{\alpha \in {{𝒫}}_n:|\mathrm{im}(\alpha )|\le r\}$ and ${𝒮}{𝒫}(n,r)=\{\alpha \in {{𝒮}{𝒫}}_n:|\mathrm{im}(\alpha )|\le r\}$ are $S(n+1,r+1)$, for $2\le r\le n-1$, and $(r+1)S(n,r+1)$, for $2\le r\le n-2$, respectively. The idempotent rank, defined as the smallest number of idempotents generating set, of the semigroup ${𝒫}(n,r)$ has the same value as the rank. Idempotent can be seen as a special case (with $p=q=1$) of $(p,q)$-potent. In this paper, we determine the $(p,q)$-potent ranks, defined as the smallest number of $(p,q)$-potents generating set, of the semigroups ${𝒫}(n,r)$, for $2\le r\le n-1$, and ${𝒮}{𝒫}(n,r)$, for $2\le r\le n-2$.

For $n\in \mathbb{N}$, let $X_n=\{a_1,a_2,\dots ,a_n\}$ be an $n$-element set and let $F=(X_n;{<}_f)$ 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 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 $PF{I}_n$ of all partial fence-preserving injections of $X_n$ and its subsemigroup $I{F}_n=\{\alpha \in PF{I}_n:{\alpha }^{-1}\in PF{I}_n\}$. Clearly, $I{F}_n$ is an inverse semigroup and contains all regular elements of $PF{I}_n.$ We characterize the Green’s relations for the semigroup $I{F}_n$. Further, we prove that the semigroup $I{F}_n$ is generated by its elements with rank $\ge n-2$. Moreover, for $n\in 2\mathbb{N}$, we find the least generating set and calculate the rank of $I{F}_n$.