Free idempotent generated semigroups: The word problem and structure via gain graphs
Igor Dolinka
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Israel Journal of Mathematics (to appear). 28 pages; some minor text overlap with arXiv:1802.02420 and arXiv:1412.5167 in the preliminary section (because of the closely related topic)
Building on the previous extensive study of Yang, Gould and the present author, we provide a more precise insight into the group-theoretical ramifications of the word problem for free idempotent generated semigroups over finite biordered sets. We prove that such word problems are in fact equivalent to the problem of computing intersections of cosets of certain subgroups of direct products of maximal subgroups of the free idempotent generated semigroup in question, thus providing decidability of those word problems under group-theoretical assumptions related to the Howson property and the coset intersection property. We also provide a basic sketch of the global semigroup-theoretical structure of an arbitrary free idempotent generated semigroup, including the characterisation of Green's relations and the key parameters of non-regular $\mathscr{D}$-classes. In particular, we prove that all Schützenberger groups of $\mathsf{IG}(\mathcal{E})$ for a finite biordered set $\mathcal{E}$ must be among the divisors of the maximal subgroups of $\mathsf{IG}(\mathcal{E})$.
An element e of an ordered semigroup $(S,\cdot,\leq)$ is called an ordered idempotent if $e\leq e^2$. We call an ordered semigroup $S$ idempotent ordered semigroup if every element of $S$ is an ordered idempotent. Every idempotent semigroup is a complete semilattice of rectangular idempotent semigroups and in this way we arrive to many other important classes of idempotent ordered semigroups.
We prove the following results: (1) Every group is a maximal subgroup of some free idempotent generated semigroup. (2) Every finitely presented group is a maximal subgroup of some free idempotent generated semigroup arising from a finite semigroup. (3) Every group is a maximal subgroup of some free regular idempotent generated semigroup. (4) Every finite group is a maximal subgroup of some free regular idempotent generated semigroup arising from a finite regular semigroup.
We use topological methods to study the maximal subgroups of the free idempotent generated semigroup on a biordered set. We use these to give an example of a free idempotent generated semigroup with maximal subgroup isomorphic to the free abelian group of rank 2. This is the first example of a non-free subgroup of a free idempotent generated semigroup.
We provide short and direct proofs for some classical theorems proved by Howie, Levi and McFadden concerning idempotent generated semigroups of transformations on a finite set.
Let T_n be the full transformation semigroup of all mappings from the set {1,...,n} to itself under composition. Let E = E(T_n) denote the set of idempotents of T_n and let e be an arbitrary idempotent satisfying |im(e)|=r < n-1. We prove that the maximal subgroup of the free idempotent generated semigroup over E containing e is isomorphic to the symmetric group S_r.
We develop some new topological tools to study maximal subgroups of free idempotent generated semigroups. As an application, we show that the rank 1 component of the free idempotent generated semigroup of the biordered set of a full matrix monoid of n x n matrices, n>2$ over a division ring Q has maximal subgroup isomorphic to the multiplicative subgroup of Q.
Given an arbitrary group $G$ we construct a semigroup of idempotents (band) $B_G$ with the property that the free idempotent generated semigroup over $B_G$ has a maximal subgroup isomorphic to $G$. If $G$ is finitely presented then $B_G$ is finite. This answers several questions from recent papers in the area.
The set of idempotents of any semigroup carries the structure of a biordered set, which contains a great deal of information concerning the idempotent generated subsemigroup of the semigroup in question. This leads to the construction of a free idempotent generated semigroup $\mathsf{IG}(\mathcal{E})$ - the `free-est' semigroup with a given biordered set $\mathcal{E}$ of idempotents. We show that when $\mathcal{E}$ is finite, the word problem for $\mathsf{IG}(\mathcal{E})$ is equivalent to a family of constraint satisfaction problems involving rational subsets of direct products of pairs of ma…
In this paper we present a new embedding of a semigroup into a semiband (idempotent-generated semigroup) of depth 4 (every element is the product of 4 idempotents) using a semidirect product construction. Our embedding does not assume that S is a monoid (although it assumes a weaker condition), and works also for (non-monoid) regular semigroups. In fact, this semidirect product is particularly useful for regular semigroups since we can defined another embedding for these semigroups into a smaller semiband of depth 2. We shall then compare our construction with other known embeddings, and we sh…
We prove that all maximal subgroups of the free idempotent generated semigroup over a band B are free for all B belonging to a band variety V if and only if V consists either of left seminormal bands, or of right seminormal bands.
A semigroup is \emph{regular} if it contains at least one idempotent in each $\mathcal{R}$-class and in each $\mathcal{L}$-class. A regular semigroup is \emph{inverse} if satisfies either of the following equivalent conditions: (i) there is a unique idempotent in each $\mathcal{R}$-class and in each $\mathcal{L}$-class, or (ii) the idempotents commute. Analogously, a semigroup is \emph{abundant} if it contains at least one idempotent in each $\mathcal{R}^*$-class and in each $\mathcal{L}^*$-class. An abundant semigroup is \emph{adequate} if its idempotents commute. In adequate semigroups, ther…
Recently, Gray and Ruskuc (arXiv:1101.1833) proved that if e is a rank k idempotent transformation of the set {1,...,n} to itself and k<=n-2, then the maximal subgroup of the free idempotent generated semigroup over the full transformation monoid T_n containing e is isomorphic to the symmetric group S_k. We prove that the same holds when T_n is replaced by PT_n, the full monoid of partial transformations on {1,...,n}.
For each group G which decomposes into a finitary direct product of free groups of finite rank we construct a regular band B such that the free idempotent generated semigroup over B contains a maximal subgroup isomorphic to G. In particular, there exists a (regular) band B_0 with the property that any idempotent generated semigroup whose biordered set is isomorphic to that of B_0 must have all its subgroups abelian.
A celebrated result of J. Thompson says that if a finite group $G$ has a fixed-point-free automorphism of prime order, then $G$ is nilpotent. The main purpose of this note is to extend this result to finite inverse semigroups. An earlier related result of B. H. Neumann says that a uniquely 2-divisible group with a fixed-point-free automorphism of order 2 is abelian. We similarly extend this result to uniquely 2-divisible inverse semigroups.
We show that the rank r component of the free idempotent generated semigroup of the biordered set of the full linear monoid of n x n matrices over a division ring Q has maximal subgroup isomorphic to the general linear group GL_r(Q), where n and r are positive integers with r < n/3.
Let $a$ be a non-invertible transformation of a finite set and let $G$ be a group of permutations on that same set. Then $\genset{G, a}\setminus G$ is a subsemigroup, consisting of all non-invertible transformations, in the semigroup generated by $G$ and $a$. Likewise, the conjugates $a^g=g^{-1}ag$ of $a$ by elements $g\in G$ generate a semigroup denoted $\genset{a^g | g\in G}$. We classify the finite permutation groups $G$ on a finite set $X$ such that the semigroups $\genset{G,a}$, $\genset{G, a}\setminus G$, and $\genset{a^g | g\in G}$ are regular for all transformations of $X$. We also cla…
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…
The category of all idempotent generated semigroups with a prescribed structure $\mathcal{E}$ of their idempotents $E$ (called the biordered set) has an initial object called the free idempotent generated semigroup over $\mathcal{E}$, defined by a presentation over alphabet $E$, and denoted by $\mathsf{IG}(\mathcal{E})$. Recently, much effort has been put into investigating the structure of semigroups of the form $\mathsf{IG}(\mathcal{E})$, especially regarding their maximal subgroups. In this paper we take these investigations in a new direction by considering the word problem for $\mathsf{IG…
We investigate monotone idempotent $n$-ary semigroups. One of the main result of this article is the generalisation of Czogala-Drewniak Theorem, which describes the idempotent monotone associative functions having neutral element. Furthermore we present the full characterisation of idempotent, monotone, $n$-associative functions on an interval which have neutral elements. %{\bl defined}. Our description provides that monotone, idempotent, $n$-ary semigroups are quasitrivial.
A regular semigroup is weakly generated by a set X if it has no proper regular subsemigroups containing X. In this paper, we study the regular semigroups weakly generated by idempotents. We show there exists a regular semigroup FI(X) weakly generated by |X| idempotents such that all other regular semigroups weakly generated by |X| idempotents are homomorphic images of FI(X). The semigroup FI(X) is defined by a presentation $\langle G(X),ρ_e\cupρ_s\rangle$ and its structure is studied. Although each of the sets $G(X)$, $ρ_e$, and $ρ_s$ is infinite for $|X|\geq 2$, we show that the word problem …
Let $X$ be a set and $\mathcal{T}_X$ be the full transformation semigroup on $X$. For a partition $\mathcal{P}$ of $X$, we consider semigroups $T(X, \mathcal{P}) = \{f\in \mathcal{T}_X| (\forall X_i\in \mathcal{P}) (\exists X_j \in \mathcal{P})\;X_i f \subseteq X_j\}$, $Σ(X, \mathcal{P}) = \{f\in T(X, \mathcal{P})|(\forall X_i \in \mathcal{P})\; Xf \cap X_i \neq \emptyset\}$, and $Γ(X, \mathcal{P}) = \{f\in \mathcal{T}_X|(\forall X_i\in \mathcal{P})(\exists X_j\in \mathcal{P})\; X_i f = X_j\}$. We characterize unit-regular elements of both $T(X, \mathcal{P})$ and $Σ(X, \mathcal{P})$ for finite…
We calculate the rank and idempotent rank of the semigroup $E(X,P)$ generated by the idempotents of the semigroup $T(X,P)$, which consists of all transformations of the finite set $X$ preserving a non-uniform partition $P$. We also classify and enumerate the idempotent generating sets of this minimal possible size. This extends results of the first two authors in the uniform case.
A semigroup is \emph{amiable} if there is exactly one idempotent in each $\mathcal{R}^*$-class and in each $\mathcal{L}^*$-class. A semigroup is \emph{adequate} if it is amiable and if its idempotents commute. We characterize adequate semigroups by showing that they are precisely those amiable semigroups which do not contain isomorphic copies of two particular nonadequate semigroups as subsemigroups.
In this paper we discuss graph inverse semigroups which are constucted from a directed graphs and study several interesting properties of graph inverse semigroups such as the nature of its idempotents, the structure of semilattice of idempotents and the like. A necessary and sufficient condition on directed graphs to have the corresponding graph inverse semigroups are primitive inverse semigroups is provided. Further the general form of elements in local submonoids of the graph inverse semigroups and the Green s relations in graph inverse semigroups in graph theoretic terms are also described
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*y=xay for x,y in S. In this article, 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 Reg(T_X^a) (consisting of all regular elements) and E_X^a (consisting of all products of idempotents), and the ideals of 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 (idem…
The idempotent problem of a finitely generated inverse semigroup is the formal language of all words over the generators representing idempotent elements. This note proves that a finitely generated inverse semigroup with regular idempotent problem is necessarily finite. This answers a question of Gilbert and Noonan Heale, and establishes a generalisation to inverse semigroups of Anisimov's Theorem for groups.
The complex algebra of an inverse semigroup with finitely many idempotents in each $\mathcal D$-class is stably finite by a result of Munn. This can be proved fairly easily using $C^*$-algebras for inverse semigroups satisfying this condition that have a Hausdorff universal groupoid, or more generally for direct limits of inverse semigroups satisfying this condition and having Hausdorff universal groupoids. It is not difficult to see that a finitely presented inverse semigroup with a non-Hausdorff universal groupoid cannot be a direct limit of inverse semigroups with Hausdorff universal groupo…
We show that the pointlike and the idempotent pointlike problems are reducible with respect to natural signatures in the following cases: the pseudovariety of all finite semigroups in which the order of every subgroup is a product of elements of a fixed set of primes; the pseudovariety of all finite semigroups in which every regular J-class is the product of a rectangular band by a group from a fixed pseudovariety of groups that is reducible for the pointlike problem, respectively graph reducible. Allowing only trivial groups, we obtain omega-reducibility of the pointlike and idempotent pointl…
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 P_n 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 have shown recently that, given a metric space $X$, the coarse equivalence classes of metrics on the two copies of $X$ form an inverse semigroup $M(X)$. Here we study the property of idempotents in $M(X)$ of being finite or infinite, which is similar to this property for projections in C*-algebras. We show that if $X$ is a free group then the unit of $M(X)$ is infinite, while if $X$ is a free abelian group then it is finite. As a by-product, we show that the inverse semigroup $M(X)$ is not a quasi-isometry invariant. More examples of finite and infinite idempotents are provided. We also giv…
We give a characterisation of the idempotents of the partition monoid, and use this to enumerate the idempotents in the finite partition, Brauer and partial Brauer monoids, giving several formulae and recursions for the number of idempotents in each monoid as well as various $\mathscr R$-, $\mathscr L$- and $\mathscr D$-classes. We also apply our results to determine the number of idempotent basis elements in the finite dimensional partition, Brauer and partial Brauer algebras.
We are interested in semigroups of the form $\langle G,a\rangle\setminus G$, where $G$ is a permutation group of degree $n$ and $a$ a non-permutation on the domain of $G$. A theorem of the first author, Mitchell and Schneider shows that, if this semigroup is idempotent-generated for all possible choices of $a$, then $G$ is the symmetric or alternating group of degree $n$, with three exceptions (having $n=5$ or $n=6$). Our purpose here is to prove stronger results where we assume that $\langle G,a\rangle\setminus G$ is idempotent-generated for all maps of fixed rank $k$. For $k\ge6$ and $n\ge2k…
The study of the free idempotent generated semigroup $\mathrm{IG}(E)$ over a biordered set $E$ began with the seminal work of Nambooripad in the 1970s and has seen a recent revival with a number of new approaches, both geometric and combinatorial. Here we study $\mathrm{IG}(E)$ in the case $E$ is the biordered set of a wreath product $G\wr \mathcal{T}_n$, where $G$ is a group and $\mathcal{T}_n$ is the full transformation monoid on $n$ elements. This wreath product is isomorphic to the endomorphism monoid of the free $G$-act $F_n(G)$ on $n$ generators, and this provides us with a convenient ap…
An algebra has the Howson property if the intersection of any two finitely generated subalgebras is finitely generated. A simple necessary and sufficient condition is given for the Howson property to hold on an inverse semigroup with finitely many idempotents. In addition, it is shown that any monogenic inverse semigroup has the Howson property.
We prove that if S is a commutative semigroup with well founded universal semilattice or a solvable inverse semigroup with well founded semilattice of idempotents, then every strongly productive ultrafilter on S is idempotent. Moreover we show that any very strongly productive ultrafilter on the free semigroup with countably many generators is sparse, answering a question of Hindman and Legette Jones.
We show that all of the Schützenberger complexes of an Adian inverse semigroup are finite if the Schützenberger complex of every positive word is finite. This enables us to solve the word problem for certain classes of Adian inverse semigroups (and hence for the corresponding Adian semigroups and Adian groups).
The study of word hyperbolic groups is a prominent topic in geometric group theory; however word hyperbolic groups are defined by a geometric condition which does not extend naturally to semigroups. We propose a linguistic definition. Roughly speaking a semigroup is word hyperbolic if its multiplication table is a context free language. For groups this definition is equivalent to the original geometric one. We also briefly consider word problems of semigroups.
Every semigroup which is a finite disjoint union of copies of the free mono- genic semigroup (natural numbers under addition) has soluble word prob- lem and soluble membership problem. Efficient algorithms are given for both problems.
A regular ordered semigroup $S$ is called right inverse if every principal left ideal of $S$ is generated by an $\mathcal{R}$-unique ordered idempotent. Here we explore the theory of right inverse ordered semigroups. We show that a regular ordered semigroup is right inverse if and only if any two right inverses of an element $a\in S$ are $\mathcal{R}$-related. Furthermore, different characterizations of right Clifford, right group-like, group like ordered semigroups are done by right inverse ordered semigroups. Thus a foundation of right inverse semigroups has been developed.
For a monoid $M$, we denote by $\mathbb G(M)$ the group of units, $\mathbb E(M)$ the submonoid generated by the idempotents, and $\mathbb G_L(M)$ and $\mathbb G_R(M)$ the submonoids consisting of all left or right units. Writing $\mathcal M$ for the (monoidal) category of monoids, $\mathbb G$, $\mathbb E$, $\mathbb G_L$ and $\mathbb G_R$ are all (monoidal) functors $\mathcal M\to\mathcal M$. There are other natural functors associated to submonoids generated by combinations of idempotents and one- or two-sided units. The above functors generate a monoid with composition as its operation. We sh…
We study algebraic and topological properties of subsemigroups of the hyperspace exp(G) of non-empty compact subsets of a topological group G endowed with the Vietoris topology and the natural semigroup operation. On this base we prove that a compact Clifford topological semigroup S is topologically isomorphic to a subsemigroup of exp(G) for a suitable topological group G if and only if S is a topological inverse semigroup with zero-dimensional idempotent semilattice.
Cross-connection theory provides the construction of a semigroup from its ideal structure using small categories. A concordant semigroup is an idempotent-connected abundant semigroup whose idempotents generate a regular subsemigroup. We characterize the categories arising from the generalised Green relations in the concordant semigroup as consistent categories and describe their interrelationship using cross-connections. Conversely, given a pair of cross-connected consistent categories, we build a concordant semigroup. We use this correspondence to prove a category equivalence between the cate…
We study the algebraic structure of the semigroup of all $2 \times 2$ tropical matrices under multiplication. Using ideas from tropical geometry, we give a complete description of Green's relations and the idempotents and maximal subgroups of this semigroup.
We associate a 2-complex to the following data: a presentation of a semigroup $S$ and a transitive action of $S$ on a set $V$ by partial transformations. The automorphism group of the action acts properly discontinuously on this 2-complex. A sufficient condition is given for the 2-complex to be simply connected. As a consequence we obtain simple topological proofs of results on presentations of Schützenberger groups. We also give a geometric proof that a finitely generated regular semigroup with finitely many idempotents has polynomial growth if and only if all its maximal subgroups are virtua…
We introduce the space function $s(n)$ of a finitely presented semigroup $S =<A\mid R>.$ To define $s(n)$ we consider pairs of words $w,w'$ over $A$ of length at most $n$ equal in $S$ and use relations from $R$ for the transformations $w=w_0\to...\to w_t= w'$; $s(n)$ bounds from above the tape space (or computer memory) sufficient to implement all such transitions $w\to...\to w'.$ One of the results obtained is the following criterion: A finitely generated semigroup $S$ has decidable word problem of polynomial space complexity if and only if $S$ is a subsemigroup of a finitely presented semigr…
This paper concerns a number of diagram categories, namely the partition, planar partition, Brauer, partial Brauer, Motzkin and Temperley-Lieb categories. If $\mathcal K$ denotes any of these categories, and if $σ\in\mathcal K_{nm}$ is a fixed morphism, then an associative operation $\star_σ$ may be defined on $\mathcal K_{mn}$ by $α\star_σβ=ασβ$. The resulting semigroup $\mathcal K_{mn}^σ=(\mathcal K_{mn},\star_σ)$ is called a sandwich semigroup. We conduct a thorough investigation of these sandwich semigroups, with an emphasis on structural and combinatorial properties such as Green's relati…
