Aperiodic Monoids Research Articles

Euclidean division by d in base b

Let b⩾2 be an integer. For each positive integer d, let Ed,b be the Euclidean division by d in base b, that is, the function which associates to a word u in {0,…,b−1}⁎, representing an integer n in base b, the unique word of the same length as u representing the quotient of the division of n by d. We describe the pure sequential transducer realizing this function and analyze the algebraic structure of its syntactic monoid. We compute its size, describe its Green's relations and its minimum ideal. As a consequence, we show that it is a group if and only if d and b are coprime numbers, it is a p-group if and only if d is a power of p and b is congruent to 1 modulo p and it is an aperiodic monoid if and only if d divides some power of b. The uniform continuity of Ed,b for the pro-group metric was studied by Reutenauer and Schützenberger in 1995. We launch a similar study for the uniform continuity of Ed,b with respect to the pro-p metric, where p is a prime number.

Limit varieties generated by finite non-J-trivial aperiodic monoids

Limit varieties generated by finite non-J-trivial aperiodic monoids

Varieties of aperiodic monoids with central idempotents whose subvariety lattice is distributive

We completely classify all varieties of aperiodic monoids with central idempotents whose subvariety lattice is distributive.

Limit varieties of aperiodic monoids with commuting idempotents

A variety of algebras is called limit if it is nonfinitely-based but all its proper subvarieties are finitely-based. A monoid is aperiodic if all its subgroups are trivial. We classify all limit varieties of aperiodic monoids with commuting idempotents.

Space complexity of reachability testing in labelled graphs

Fix an algebraic structure (A,⁎). Given a graph G=(V,E) and the labelling function ϕ (ϕ:E→A) for the edges, two nodes s, t∈V, and a subset F⊆A, the A-Reach problem asks if there is a path p (need not be simple) from s to t whose yield (result of the operation in the ordered set of the labels of the edges constituting the path) is in F. On the complexity frontier of this problem, we show the following results.•When A is a group whose size is polynomially bounded in the size of the graph (hence equivalently presented as a multiplication table at the input), and the graph is undirected, the A-Reach problem is in L. Building on this, using a decomposition in [4], we show that, when A is a fixed quasi-group, and the graph is undirected, the A-Reach problem is in L. In a contrast, we show NL-hardness of the problem over bidirected graphs, when A is a matrix group over Q, or an aperiodic monoid.•As our main theorem, we prove a dichotomy for graphs labelled with fixed aperiodic monoids by showing that for every fixed aperiodic monoid A, A-Reach problem is either in L or is NL-complete.•We show that there exists a monoid M, such that the reachability problem in general DAGs can be reduced to A-Reach problem for planar non-bipartite DAGs labelled with M. In contrast, we show that if the planar DAGs that we obtain above are bipartite, the problem can be further reduced to reachability testing in planar DAGs and hence is in UL.

On uniformly continuous functions for some profinite topologies

Given a variety of finite monoids V, a subset of a monoid is a V-subset if its syntactic monoid belongs to V. A function between two monoids is V-preserving if it preserves V-subsets under preimages and it is hereditary V-preserving if it is W-preserving for every subvariety W of V. The aim of this paper is to study hereditary V-preserving functions when V is one of the following varieties of finite monoids: groups, p-groups, aperiodic monoids, commutative monoids and all monoids.

A SUFFICIENT CONDITION UNDER WHICH A SEMIGROUP IS NONFINITELY BASED

We give a sufficient condition under which a semigroup is nonfinitely based. As an application, we show that a certain variety is nonfinitely based, and we indicate the additional analysis (to be presented in a forthcoming paper), which shows that this example is a new limit variety of aperiodic monoids.

Inherently Non-Finitely Generated Varieties of Aperiodic Monoids with Central Idempotents

Let denote the class of aperiodic monoids with central idempotents. A subvariety of that is not contained in any finitely generated subvariety of is said to be inherently non-finitely generated. A characterization of inherently non-finitely generated subvarieties of , based on identities that they cannot satisfy and monoids that they must contain, is given. It turns out that there exists a unique minimal inherently non-finitely generated subvariety of , the inclusion of which is both necessary and sufficient for a subvariety of to be inherently non-finitely generated. Further, it is decidable in polynomial time if a finite set of identities defines an inherently nonfinitely generated subvariety of .

Finitely based monoids

We present a method for proving that a semigroup is finitely based (FB) and find some new sufficient conditions under which a monoid is FB. As an application, we find a class of finite aperiodic monoids where the finite basis property behaves in a complicated way with respect to the lattice operations but can be recognized by a simple algorithm. The method results in a short proof of the theorem of E. Lee that every monoid that satisfies $$xt_1xyt_2y \approx xt_1yxt_2y$$ and $$xyt_1xt_2y \approx yxt_1xt_2y$$ is FB. Also, the method gives an alternative proof of the theorem of F. Blanchet-Sadri that a pseudovariety of $$n$$ -testable languages is FB if and only if $$n \le 3$$ .

Non-finitely based monoids

We present a general method for proving that a semigroup is non-finitely based (NFB). The method is strong enough to cover a majority of existing non-finite basis arguments for a periodic semigroups and allows to generalize several previous results and to simplify their proofs. The method also allows to remove one of the requirements on the “special system of identities” used by Perkins in 1968 to find the first two examples of finite NFB semigroups. We use our method to prove eleven new sufficient conditions under which a monoid is NFB. As an application, we find infinitely many new examples of finite finitely based aperiodic monoids whose direct product is NFB.

On Certain Cross Varieties of Aperiodic Monoids with Commuting Idempotents

Let \({\fancyscript{A}_\mathrm{com}}\) denote the class of aperiodic monoids with commuting idempotents. It is shown that any subvariety of \({\fancyscript{A}_\mathrm{com}}\) that satisfies the identity \({x^2yx \approx xyx^2}\) is Cross if and only if it excludes the almost Cross varieties \({\mathbf{J_1}, \mathbf{J_2}}\), and \({\mathbf{L}}\). Consequently, these three almost Cross varieties are unique among all subvarieties of \({\fancyscript{A}_\mathrm{com}}\) that satisfy the identity \({x^2yx \approx xyx^2}\). On the other hand, the existence of almost Cross subvarieties of \({\fancyscript{A}_\mathrm{com}}\) different from \({\mathbf{J_1}, \mathbf{J_2}}\), and \({\mathbf{L}}\) is also established.

Hardness of learning loops, monoids, and semirings

We show that each randomized o(|G|2)-query algorithm can recover only an expected o(1) fraction of the Cayley table of some finite Abelian loop (G,⋅), where both multiplication and inversion queries are allowed. Furthermore, each randomized o(|R|2)-query algorithm can recover only an expected o(1) fraction of any of the Cayley tables of some finite commutative semiring (R,+,⋅), with (R,+) being a commutative aperiodic monoid, where each query may ask for x+y or x⋅y for any x, y∈R.

Existence of a new limit variety of aperiodic monoids

Presently, the two limit varieties of Marcel Jackson are the only known examples of limit varieties of aperiodic monoids. This article establishes the existence of a new example by showing that a certain aperiodic monoid of order seven is non-finitely based. This monoid turns out to be a smallest non-finitely based monoid that is not inherently non-finitely based.

Almost Cross varieties of aperiodic monoids with central idempotents

It is known that the class \({\fancyscript{A}}\) of aperiodic monoids with central idempotents contains precisely two finitely generated, almost Cross subvarieties. The present article exhibits the first example of a non-finitely generated, almost Cross subvariety of \({\fancyscript{A}}\) and verifies that it is in fact unique. Consequently, the class \({\fancyscript{A}}\) contains precisely three almost Cross subvarieties.

On free spectra of finite monoids from the pseudovariety DA

The pseudovariety DA consists of all aperiodic finite monoids all of whose regular Open image in new window-classes are subsemigroups (that is, rectangular subbands); this pseudovariety appears quite frequently in various contexts in finite semigroup theory. In this note we prove that all its members have a log-polynomial free spectrum, thereby making a new step towards proving the Seif conjecture on the dichotomy of free spectra of finite monoids.

Cross varieties of aperiodic monoids with central idempotents

Let \boldsymbol A denote the class of all aperiodic monoids with central idempotents. A description of all Cross subvarieties of \boldsymbol A , based on identities that they satisfy and monoids that they cannot contain, is given. The two limit subvarieties of \boldsymbol A , published by Marcel Jackson in 2005, turn out to be the only finitely generated, almost Cross subvarieties of \boldsymbol A . It follows that it is decidable in quartic time if a finite monoid in \boldsymbol A generates a Cross variety.

Pseudovarieties generated by Brauer type monoids

It is proved that the series of all Brauer monoids generates the pseudovariety of all finite monoids while the series of their aperiodic analogues, the Jones monoids (also called Temperly–Lieb monoids), generates the pseudovariety of all finite aperiodic monoids. The proof is based on the analysis of wreath product decomposition and Krohn–Rhodes theory. The fact that the Jones monoids form a generating series for the pseudovariety of all finite aperiodic monoids can be viewed as solution of an old problem popularized by J.-É. Pin. For the latter, the relationship between the Jones monoids and the monoids of order preserving mappings of a chain of length n is investigated.

A PROFINITE APPROACH TO STABLE PAIRS

We give a short proof, using profinite techniques, that idempotent pointlikes, stable pairs and triples are decidable for the pseudovariety of aperiodic monoids. Stable pairs are also described for the pseudovariety of all finite monoids.

FINITELY GENERATED LIMIT VARIETIES OF APERIODIC MONOIDS WITH CENTRAL IDEMPOTENTS

A non-finitely based variety of algebras is said to be a limit variety if all its proper subvarieties are finitely based. Recently, Marcel Jackson published two examples of finitely generated limit varieties of aperiodic monoids with central idempotents and questioned whether or not they are unique. The present article answers this question affirmatively.

A SURVEY ON SMALL FRAGMENTS OF FIRST-ORDER LOGIC OVER FINITE WORDS

We consider fragments of first-order logic over finite words. In particular, we deal with first-order logic with a restricted number of variables and with the lower levels of the alternation hierarchy. We use the algebraic approach to show decidability of expressibility within these fragments. As a byproduct, we survey several characterizations of the respective fragments. We give complete proofs for all characterizations and we provide all necessary background. Some of the proofs seem to be new and simpler than those which can be found elsewhere. We also give a proof of Simon's theorem on factorization forests restricted to aperiodic monoids because this is simpler and sufficient for our purpose.