Fast Fourier Transforms for Finite Inverse Semigroups
Martin Malandro
备注：
ver 3: Added improved upper and lower bounds for the memory required by the fast zeta transform on the rook monoid. ver 2: Corrected typos and (naive) bounds on memory requirements. 30 pages, 0 figures
We extend the theory of fast Fourier transforms on finite groups to finite inverse semigroups. We use a general method for constructing the irreducible representations of a finite inverse semigroup to reduce the problem of computing its Fourier transform to the problems of computing Fourier transforms on its maximal subgroups and a fast zeta transform on its poset structure. We then exhibit explicit fast algorithms for particular inverse semigroups of interest--specifically, for the rook monoid and its wreath products by arbitrary finite groups.
We continue the study of permutations of a finite regular semigroup that map each element to one of its inverses, providing a complete description in the case of semigroups whose idempotent generated subsemigroup is a union of groups. We show, in two ways, how to construct an involution matching on the semigroup of all transformations which either preserve or reverse orientation of a finite cycle. Finally, by way of application, we prove that finite full transformation semigroups have no cover by inverse semigroups that is closed under intersection.
We describe general methods for enumerating subsemigroups of finite semigroups and techniques to improve the algorithmic efficiency of the calculations. As a particular application we use our algorithms to enumerate all transformation semigroups up to degree 4. Classification of these semigroups up to conjugacy, isomorphism and anti-isomorphism, by size and rank, provides a solid base for further investigations of transformation semigroups.
Let $a$ be an element of a semigroup $S$. The local subsemigroup of $S$ with respect to $a$ is the subsemigroup $aSa$ of $S$. The variant of $S$ with respect to $a$ is the semigroup with underlying set $S$ and operation $\star_a$ defined by $x\star_ay=xay$ for $x,y\in S$. We show that the following classes contain precisely the same semigroups, up to isomorphism: all local subsemigroups of all finite full transformation semigroups; and all variants of all finite full transformation semigroups. This result was discovered as a result of some experiments (and accidents) when working with the Semi…
Let $Ω$ be a finite set and $T(Ω)$ be the full transformation monoid on $Ω$. The rank of a transformation $t\in T(Ω)$ is the natural number $|Ωt|$. Given $A\subseteq T(Ω)$, denote by $\langle A\rangle$ the semigroup generated by $A$. Let $k$ be a fixed natural number such that $2\le k\le |Ω|$. In the first part of this paper we (almost) classify the permutation groups $G$ on $Ω$ such that for all rank $k$ transformation $t\in T(Ω)$, every element in $S_t:=\langle G,t\rangle$ can be written as a product $eg$, where $e^2=e\in S_t$ and $g\in G$. In the second part we prove, among other results, t…
Let $X$ be a nonempty set, and let $\mathcal{T}_X$ be the full transformation semigroup on $X$. For a partition $\mathcal{P} = \{X_i \;|\; i\in I\}$ of $X$, we consider the semigroup $T(X, \mathcal{P}) = \{f\in \mathcal{T}_X\;|\; \forall X_i\;\exists X_j,\; X_i f \subseteq X_j\}$, the subsemigroup $Σ(X, \mathcal{P}) = \{f\in T(X, \mathcal{P})\;|\; Xf \cap X_i \neq \emptyset\; \forall X_i\}$, and the group of units $S(X, \mathcal{P})$ of $T(X, \mathcal{P})$. In this paper, we first characterize the elements of $Σ(X, \mathcal{P})$. For a permutation $f$ of finite $X$, we next observe whether the…
With each semigroup one can associate a partial algebra, called the biordered set, which captures important algebraic and geometric features of the structure of idempotents of that semigroup. For a biordered set $\mathcal{E}$, one can construct the free idempotent-generated semigroup over $\mathcal{E}$, $\mathsf{IG}(\mathcal{E})$, which is the free-est semigroup (in a definite categorical sense) whose biorder of idempotents is isomorphic to $\mathcal{E}$. Studies of these intriguing objects have been recently focusing on their particular aspects, such as maximal subgroups, the word problem, et…
Tetris is a popular puzzle video game, invented in 1984. We formulate two versions of the game as a transformation semigroup and use this formulation to view the game through the lens of Krohn-Rhodes theory. In a variation of the game upon which it restarts if the player loses, we find permutation group structures, including the symmetric group $S_5$ which contains a non-abelian simple group as a subgroup. This implies, at least in a simple case, that iterated Tetris is finitarily computationally universal.
Let $X$ be a nonempty set and let $T(X)$ be the full transformation semigroup on $X$. The main objective of this paper is to study the subsemigroup $\overlineΩ(X, Y)$ of $T(X)$ defined by \[\overlineΩ(X, Y) = \{f\in T(X)\colon Yf = Y\},\] where $Y$ is a fixed nonempty subset of $X$. We describe regular elements in $\overlineΩ(X, Y)$ and show that $\overlineΩ(X, Y)$ is regular if and only if $Y$ is finite. We characterize unit-regular elements in $\overlineΩ(X, Y)$ and prove that $\overlineΩ(X, Y)$ is unit-regular if and only if $X$ is finite. We characterize Green's relations on $\overlineΩ(X,…
Motivated by a problem on the dynamics of compositions of plane hyperbolic isometries, we prove several fundamental results on semigroups of isometries, thought of as real Möbius transformations. We define a semigroup $S$ of Möbius transformations to be \emph{semidiscrete} if the identity transformation is not an accumulation point of $S$. We say that $S$ is \emph{inverse free} if it does not contain the identity element. One of our main results states that if $S$ is a semigroup generated by some finite collection $\mathcal{F}$ of Möbius transformations, then $S$ is semidiscrete and inverse fr…
Let $X$ be a finite set such that $|X|=n$, and let $k< n/2$. A group is $k$-homogeneous if it has only one orbit on the sets of size $k$. The aim of this paper is to prove some general results on permutation groups and then apply them to transformation semigroups. On groups we find the minimum number of permutations needed to generate $k$-homogeneous groups (for $k\ge 1$); in particular we show that $2$-homogeneous groups are $2$-generated. We also describe the orbits of $k$-homogenous groups on partitions with $n-k$ parts, classify the $3$-homogeneous groups $G$ whose orbits on $(n-3)$-partit…
We formulate and explain the extended Burrows-Wheeler transform of Mantaci et al from the viewpoint of permutations on a chain taken as a union of partial order-preserving mappings. In so doing we establish a link with syntactic semigroups of languages that are themselves cyclic semigroups. We apply the extended transform with a view to generating de Bruijn words through inverting the transform.
The words separation problem, originally formulated by Goralcik and Koubek (1986), is stated as follows. Let $Sep(n)$ be the minimum number such that for any two words of length $\le n$ there is a deterministic finite automaton with $Sep(n)$ states, accepting exactly one of them. The problem is to find the asymptotics of the function $Sep$. This problem is inverse to finding the asymptotics of the length of the shortest identity in full transformation semigroups $T_k$. The known lower bound on $Sep$ stems from the unary identity in $T_k$. We find the first series of identities in $T_k$ which a…
