## Publications

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Book Chapter
Cain, Alan J., and António Malheiro. "Combinatorics of Cyclic Shifts in Plactic, Hypoplactic, Sylvester, and Related Monoids." Combinatorics on Words: 11th International Conference, WORDS 2017, Montréal, QC, Canada, September 11-15, 2017, Proceedings. Eds. Srečko Brlek, Francesco Dolce, Christophe Reutenauer, and Élise Vandomme. Cham: Springer International Publishing, 2017. 190-202. Abstract

The cyclic shift graph of a monoid is the graph whose vertices are elements of the monoid and whose edges link elements that differ by a cyclic shift. For certain monoids connected with combinatorics, such as the plactic monoid (the monoid of Young tableaux) and the sylvester monoid (the monoid of binary search trees), connected components consist of elements that have the same evaluation (that is, contain the same number of each generating symbol). This paper discusses new results on the diameters of connected components of the cyclic shift graphs of the finite-rank analogues of these monoids, showing that the maximum diameter of a connected component is dependent only on the rank. The proof techniques are explained in the case of the sylvester monoid.

Conference Paper
Malheiro, A. "On Finite Semigroup Cross-Sections and Complete Rewriting Systems." International Conference on Theoretical and Mathematical Foundations of Computer Science, TMFCS-08, Orlando, Florida, USA, July 7-10, 2008. 2008. 59-63. Abstract

In this paper we obtain a [finite] complete rewriting system defining a semigroup/monoid S, from a given finite
right cross-section of a subsemigroup/submonoid defined by a [finite] complete presentation. In the semigroup case the subsemigroup must have a right identity element which must also be part of the cross-section. In the monoid case the submonoid and the cross-section must include the identity of the semigroup. The result on semigroups allow us to show that if G is a group defined by a [finite] complete rewriting system then the completely simple semigroup M[G; I, J; P] is also defined by a [finite] complete rewriting system.

Conference Proceedings
Malheiro, A. On trivializers and subsemigroups.. Semigroups and formal languages. Proceedings of the international conference in honour of the 65th birthday of Donald B. McAlister. Lisboa, Portugal, July 12–15, 2005.: Hackensack, NJ: World Scientific, 2007. Abstract

The aim of this paper is to develop the calculus of trivializers for subsemigroups. Given a finite presentation defining a semigroup S and a trivializer of the Squier complex of , we obtain an infinite trivializer of the Squier complex of a finite presentation defining a subsemigroup of S. Also, we give a method to find finite trivializers for special subsemigroups and hence to show that those subsemigroups have finite derivation type (FDT). An application of this method is given: we prove that if is a band of monoids having FDT, then so does Sα, for any α ∈Y.

Journal Article
Araújo, J., M. Kinyon, and A. Malheiro. "A characterization of adequate semigroups by forbidden subsemigroups." Proc. R. Soc. Edinb., Sect. A, Math.. 143 (2013): 1115-1122. AbstractWebsite

A semigroup is amiable if there is exactly one idempotent in each ℛ*-class and in each ℒ*-class. A semigroup is adequate if it is amiable and if its idempotents commute. We characterize adequate semigroups by showing that they are precisely those amiable semigroups that do not contain isomorphic copies of two particular non-adequate semigroups as subsemigroups.

Cain, Alan J., António Malheiro, and Fábio M. Silva. "Combinatorics of patience sorting monoids." (Submitted). Abstract

This paper makes a combinatorial study of the two monoids and the two types of tableaux that arise from the two possible generalizations of the Patience Sorting algorithm from permutations (or standard words) to words. For both types of tableaux, we present Robinson--Schensted--Knuth-type correspondences (that is, bijective correspondences between word arrays and certain pairs of semistandard tableaux of the same shape), generalizing two known correspondences: a bijective correspondence between standard words and certain pairs of standard tableaux, and an injective correspondence between words and pairs of tableaux.

We also exhibit formulas to count both the number of each type of tableaux with given evaluations (that is, containing a given number of each symbol). Observing that for any natural number $n$, the $n$-th Bell number is given by the number of standard tableaux containing $n$ symbols, we restrict the previous formulas to standard words and extract a formula for the Bell numbers. Finally, we present a hook length formula' that gives the number of standard tableaux of a given shape and deduce some consequences.

Malheiro, A. "Complete rewriting systems for codified submonoids." Int. J. Algebra Comput.. 15 (2005): 207-216. AbstractWebsite

Given a complete rewriting system R on X and a subset X0 of X+ satisfying certain conditions, we present a complete rewriting system for the submonoid of M(X;R) generated by X0. The obtained result will be applied to the group of units of a monoid satisfying H1 = D1. On the other hand we prove that all maximal subgroups of a monoid defined by a special rewriting system are isomorphic.

Cain, A. J., António Malheiro, and Fábio M. Silva. "Conjugacy in Patience Sorting monoids." (Submitted). Abstract

The cyclic shift graph of a monoid is the graph whose vertices are the elements of the monoid and whose edges connect elements that are cyclic shift related. The Patience Sorting algorithm admits two generalizations to words, from which two kinds of monoids arise, the $\belr$ monoid and the $\bell$ (also known as Bell) monoid. Like other monoids arising from combinatorial objects such as the plactic and the sylvester, the connected components of the cyclic shift graph of the $\belr$ monoid consists of elements that have the same number of each of its composing symbols. In this paper, with the aid of the computational tool SageMath, we study the diameter of the connected components from the cyclic shift graph of the $\belr$ monoid.

Within the theory of monoids, the cyclic shift relation, among other relations, generalizes the relation of conjugacy for groups. We examine several of these relations for both the $\belr$ and the $\bell$ monoids.

Araújo, J., J. Konieczny, and A. Malheiro. "Conjugation in semigroups." J. Algebra. 403 (2014): 93-134. AbstractWebsite

The action of any group on itself by conjugation and the corresponding conjugacy relation play an important role in group theory. There have been several attempts to extend the notion of conjugacy to semigroups. In this paper, we present a new definition of conjugacy that can be applied to an arbitrary semigroup and it does not reduce to the universal relation in semigroups with a zero. We compare the new notion of conjugacy with existing definitions, characterize the conjugacy in various semigroups of transformations on a set, and count the number of conjugacy classes in these semigroups when the set is infinite.

Cain, A. J., and A. Malheiro. "Crystallizing the hypoplactic monoid: from quasi-Kashiwara operators to the Robinson--Schensted-type correspondence for quasi-ribbon tableaux." Journal of Algebraic Combinatorics. 45.2 (2017): 475-524. AbstractWebsite

Crystal graphs, in the sense of Kashiwara, carry a natural monoid structure given by identifying words labelling vertices that appear in the same position of isomorphic components of the crystal. In the particular case of the crystal graph for the q-analogue of the special linear Lie algebra sln, this monoid is the celebrated plactic monoid, whose elements can be identified with Young tableaux. The crystal graph and the so-called Kashiwara operators interact beautifully with the combinatorics of Young tableaux and with the Robinson--Schensted correspondence and so provide powerful combinatorial tools to work with them. This paper constructs an analogous quasi-crystal' structure for the hypoplactic monoid, whose elements can be identified with quasi-ribbon tableaux and whose connection with the theory of quasi-symmetric functions echoes the connection of the plactic monoid with the theory of symmetric functions. This quasi-crystal structure and the associated quasi-Kashiwara operators are shown to interact just as neatly with the combinatorics of quasi-ribbon tableaux and with the hypoplactic version of the Robinson--Schensted correspondence. A study is then made of the interaction of the crystal graph of the plactic monoid and the quasi-crystal graph for the hypoplactic monoid. Finally, the quasi-crystal structure is applied to prove some new results about the hypoplactic monoid.

Cain, Alan J., and António Malheiro. "Crystals and trees: Quasi-Kashiwara operators, monoids of binary trees, and Robinson–Schensted-type correspondences." Journal of Algebra. 502 (2018): 347-381. AbstractWebsite

Kashiwara's crystal graphs have a natural monoid structure that arises by identifying words labelling vertices that appear in the same position of isomorphic components. The celebrated plactic monoid (the monoid of Young tableaux), arises in this way from the crystal graph for the q-analogue of the general linear Lie algebra gln, and the so-called Kashiwara operators interact beautifully with the combinatorics of Young tableaux and with the Robinson–Schensted–Knuth correspondence. The authors previously constructed an analogous ‘quasi-crystal’ structure for the related hypoplactic monoid (the monoid of quasi-ribbon tableaux), which has similarly neat combinatorial properties. This paper constructs an analogous ‘crystal-type’ structure for the sylvester and Baxter monoids (the monoids of binary search trees and pairs of twin binary search trees, respectively). Both monoids are shown to arise from this structure just as the plactic monoid does from the usual crystal graph. The interaction of the structure with the sylvester and Baxter versions of the Robinson–Schensted–Knuth correspondence is studied. The structure is then applied to prove results on the number of factorizations of elements of these monoids, and to prove that both monoids satisfy non-trivial identities.

Araújo, J., M. Kinyon, J. Konieczny, and A. Malheiro. "Decidability and Independence of Conjugacy Problems in Finitely Presented Monoids." (Submitted). Abstract

There have been several attempts to extend the notion of conjugacy from groups to monoids.
The aim of this paper is study the decidability and independence of conjugacy problems
for three of these notions (which we will denote by $\sim_p$, $\sim_o$, and $\sim_c$) in
certain classes of finitely presented monoids. We will show that in the class of polycyclic monoids,
$p$-conjugacy is almost'' transitive, $\sim_c$ is strictly included in $\sim_p$, and
the $p$- and $c$-conjugacy problems are decidable with linear compexity.
For other classes of monoids, the situation is more complicated.
We show that there exists a monoid $M$ defined by a finite complete
presentation such that the $c$-conjugacy problem for $M$ is undecidable, and
that for finitely presented monoids, the $c$-conjugacy problem and the word
problem are independent, as are the $c$-conjugacy and $p$-conjugacy problems.

Cain, A. J., and A. Malheiro. "Deciding conjugacy in sylvester monoids and other homogeneous monoids." Int. J. Algebra Comput.. 25 (2015): 899-915. AbstractWebsite

We give a combinatorial characterization of conjugacy in the sylvester monoid, showing that conjugacy is decidable for this monoid. We then prove that conjugacy is undecidable in general for homogeneous monoids and even for multihomogeneous monoids.

Gray, R. D., and A. Malheiro. "Finite complete rewriting systems for regular semigroups." Theor. Comput. Sci.. 412 (2011): 654-661. AbstractWebsite

It is proved that, given a (von Neumann) regular semigroup with finitely many left and right ideals, if every maximal subgroup is presentable by a finite complete rewriting system, then so is the semigroup. To achieve this, the following two results are proved: the property of being defined by a finite complete rewriting system is preserved when taking an ideal extension by a semigroup defined by a finite complete rewriting system; a completely 0-simple semigroup with finitely many left and right ideals admits a presentation by a finite complete rewriting system provided all of its maximal subgroups do.

Malheiro, A. "Finite derivation type for large ideals." Semigroup Forum. 78 (2009): 450-485. AbstractWebsite

n this paper we give a partial answer to the following question: does a large subsemigroup of a semigroup S with the finite combinatorial property finite derivation type (FDT) also have the same property? A positive answer is given for large ideals. As a consequence of this statement we prove that, given a finitely presented Rees matrix semigroup M[S;I,J;P], the semigroup S has FDT if and only if so does M[S;I,J;P].

Malheiro, A. "Finite derivation type for Rees matrix semigroups." Theor. Comput. Sci.. 355 (2006): 274-290. AbstractWebsite

This paper introduces the topological finiteness condition finite derivation type (FDT) on the class of semigroups. This notion is naturally extended from the monoid case. With this new concept we are able to prove that if a Rees matrix semigroup M[S;I,J;P] has FDT then the semigroup S also has FDT. Given a monoid S and a finitely presented Rees matrix semigroup M[S;I,J;P] we prove that if the ideal of S generated by the entries of P has FDT, then so does M[S;I,J;P]. In particular, we show that, for a finitely presented completely simple semigroup M, the Rees matrix semigroup M=M[S;I,J;P] has FDT if and only if the group S has FDT.

Malheiro, A. "Finite derivation type for semilattices of semigroups." Semigroup Forum. 84 (2012): 515-526. AbstractWebsite

In this paper we investigate how the combinatorial property finite derivation type (FDT) is preserved in a semilattice of semigroups. We prove that if S=S[Y,S_α] is a semilattice of semigroups such that Y is finite and each S_α (α∈Y) has FDT, then S has FDT. As a consequence we can show that a strong semilattice of semigroups S[Y,S_α,λ_{α,β}] has FDT if and only if Y is finite and every semigroup S α (α∈Y) has FDT.

Cain, A. J., R. D. Gray, and A. Malheiro. "Finite Gröbner-Shirshov bases for plactic algebras and biautomatic structures for plactic monoids." J. Algebra. 423 (2015): 37-53. AbstractWebsite

This paper shows that every Plactic algebra of finite rank admits a finite Gröbner--Shirshov basis. The result is proved by using the combinatorial properties of Young tableaux to construct a finite complete rewriting system for the corresponding Plactic monoid, which also yields the corollaries that Plactic monoids of finite rank have finite derivation type and satisfy the homological finiteness properties left and right $\mathrm{FP}_\infty$. Also, answering a question of Zelmanov, we apply this rewriting system and other techniques to show that Plactic monoids of finite rank are biautomatic.

Araújo, João, Michael Kinyon, Janusz Konieczny, and António Malheiro. "Four notions of conjugacy for abstract semigroups." Proceedings of the Royal Society of Edinburgh: Section A Mathematics. 147 (2017): 1169-1214. AbstractWebsite

n/a

Gray, R. D., A. Malheiro, and S. J. Pride. "Homotopy bases and finite derivation type for Schützenberger groups of monoids." J. Symb. Comput.. 50 (2013): 50-78. AbstractWebsite

Given a finitely presented monoid and a homotopy base for the monoid, and given an arbitrary Schutzenberger group of the monoid, the main result of this paper gives a homotopy base, and presentation, for the Schutzenberger group. In the case that the R-class R' of the Schutzenberger group G(H) has only finitely many H-classes, and there is an element s of the multiplicative right pointwise stabilizer of H, such that under the left action of the monoid on its R-classes the intersection of the orbit of the R-class of s with the inverse orbit of R' is finite, then finiteness of the presentation and of the homotopy base is preserved.

Gray, R. D., and A. Malheiro. "Homotopy bases and finite derivation type for subgroups of monoids." J. Algebra. 410 (2014): 53-84. AbstractWebsite

Given a monoid defined by a presentation, and a homotopy base for the derivation graph associated to the presentation, and given an arbitrary subgroup of the monoid, we give a homotopy base (and presentation) for the subgroup. If the monoid has finite derivation type (FDT), and if under the action of the monoid on its subsets by right multiplication the strong orbit of the subgroup is finite, then we obtain a finite homotopy base for the subgroup, and hence the subgroup has FDT. As an application we prove that a regular monoid with finitely many left and right ideals has FDT if and only if all of its maximal subgroups have FDT. We use this to show that a finitely presented regular monoid with finitely many left and right ideals satisfies the homological finiteness condition FP3 if all of its maximal subgroups satisfy the condition FP_3.

Cain, A. J., A. Malheiro, and F. M. Silva. "The monoids of the patience sorting algorithm." (Submitted). Abstract

The left patience sorting (lPS) monoid, also known in the literature as the Bell monoid, and the right patient sorting (rPS) monoid are introduced by defining certain congruences on words. Such congruences are constructed using insertion algorithms based on the concept of decreasing subsequences.
Presentations for these monoids are given.

Each finite-rank rPS monoid is shown to have polynomial growth and to satisfy a non-trivial identity (dependent on its rank), while the infinite rank rPS monoid does not satisfy a non-trivial identity. The lPS monoids of finite rank have exponential growth and thus do not satisfy non-trivial identities. The
complexity of the insertion algorithms is discussed.

rPS monoids of finite rank are shown to be automatic and to have recursive complete presentations. When the rank is $1$ or $2$, they are also biautomatic. lPS monoids of finite rank are shown to have finite complete presentations and to be biautomatic.

Araújo, J., and A. Malheiro. "On finite complete presentations and exact decompositions of semigroups." Commun. Algebra. 39 (2011): 3866-3878. AbstractWebsite

We prove that given a finite (zero) exact right decomposition (M, T) of a semigroup S, if M is defined by a finite complete presentation, then S is also defined by a finite complete presentation. Exact right decompositions are natural generalizations to semigroups of coset decompositions in groups. As a consequence, we deduce that any Zappa–Szép extension of a monoid defined by a finite complete presentation, by a finite monoid, is also defined by such a presentation.

It is also proved that a semigroup M^0[A; I, J; P], where A and P satisfy some very general conditions, is also defined by a finite complete presentation.

Cain, A. J., R. D. Gray, and A. Malheiro. "On finite complete rewriting systems, finite derivation type, and automaticity for homogeneous monoids." Information and Computation. 255 (2017): 68-93. AbstractWebsite

The class of finitely presented monoids defined by homogeneous (length-preserving) relations
is considered. The properties of admitting a finite complete rewriting system, having finite derivation type, being automatic, and being biautomatic, are investigated for monoids in this class. The first main result shows that for any possible combination of these properties and their negations there is a homoegenous monoid with exactly this combination of properties. We then extend this result to show that the same statement holds even if one restricts attention to the class of $n$-ary multihomogeneous monoids (meaning every side of every relation has fixed length $n$, and all relations are also content preserving).

Gray, R. D., A. Malheiro, and S. J. Pride. "On properties not inherited by monoids from their Schützenberger groups." Inf. Comput.. 209 (2011): 1120-1134. AbstractWebsite

We give an example of a monoid with finitely many left and right ideals, all of whose Schützenberger groups are presentable by finite complete rewriting systems, and so each have finite derivation type, but such that the monoid itself does not have finite derivation type, and therefore does not admit a presentation by a finite complete rewriting system. The example also serves as a counterexample to several other natural questions regarding complete rewriting systems and finite derivation type. Specifically it allows us to construct two finitely generated monoids M and N with isometric Cayley graphs, where N has finite derivation type (respectively, admits a presentation by a finite complete rewriting system) but M does not. This contrasts with the case of finitely generated groups for which finite derivation type is known to be a quasi-isometry invariant. The same example is also used to show that neither of these two properties is preserved under finite Green index extensions.

Cain, A. J., R. D. Gray, and A. Malheiro. "Rewriting systems and biautomatic structures for Chinese, hypoplactic, and sylvester monoids." Int. J. Algebra Comput.. 25 (2015): 51-80. AbstractWebsite

This paper studies complete rewriting systems and biautomaticity for three interesting classes of finite-rank homogeneous monoids: Chinese monoids, hypoplactic monoids, and sylvester monoids. For Chinese monoids, we first give new presentations via finite complete rewriting systems, using more lucid constructions and proofs than those given independently by Chen & Qui and Güzel Karpuz; we then construct biautomatic structures. For hypoplactic monoids, we construct finite complete rewriting systems and biautomatic structures. For sylvester monoids, which are not finitely presented, we prove that the standard presentation is an infinite complete rewriting system, and construct biautomatic structures. Consequently, the monoid algebras corresponding to monoids of these classes are automaton algebras in the sense of Ufnarovskij.