António Malheiro

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.

We exhibit faithful representations of the hypoplactic, stalactic, taiga, sylvester, Baxter and right patience sorting monoids of each finite rank as monoids of upper triangular matrices over any semiring from a large class including the tropical semiring and fields of characteristic 0. By analysing the image of these representations, we show that the variety generated by a single hypoplactic (respectively, stalactic or taiga) monoid of rank at least 2 coincides with the variety generated by the natural numbers together with a fixed finite monoid (respectively, F) and forms a proper subvariety of the variety generated by the plactic monoid of rank 2.

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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.

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.

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.

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).

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.

This paper uses the combinatorics of Young tableaux to prove the plactic monoid of infinite rank does not satisfy a non-trivial identity, by showing that the plactic monoid of rank n cannot satisfy a non-trivial identity of length less than or equal to n. A new identity is then proven to hold for the monoid of n×n upper-triangular tropical matrices. Finally, a straightforward embedding is exhibited of the plactic monoid of rank 3 into the direct product of two copies of the monoid of 3×3 upper-triangular tropical matrices, giving a new proof that the plactic monoid of rank 3 satisfies a non-trivial identity.

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.

This paper considers whether non-trivial identities are satisfied by certain ‘plactic-like’ monoids that, like the plactic monoid, are closely connected with combinatorics. New results show that the hypoplactic, sylvester, Baxter, stalactic, and taiga monoids satisfy identities. The existing state of knowledge is discussed for the plactic and Bell monoids.

In 1900, Hilbert gave a lecture at the International Congress of Mathematicians in Paris, for which he prepared 23 problems that mathematicians should solve during the twentieth century. It was found that there was a note on a 24th Problem focusing on the problem of simplicity of proofs. One of the lines of research that was generated from this problem was the identification of proofs. In this article, we present a possible method for exploring the identification of proofs based on the membership problem original from the theory of polynomial rings. To show this, we start by giving a complete worked-out example of a membership problem, that is, the problem of checking if a given polynomial belongs to an ideal generated by finitely many polynomials. This problem can be solved by considering Gröbner bases and the corresponding reductions. Each reduction is a simplification of the polynomial and it corresponds to a rewriting step. In proving that a polynomial is a member of an ideal, a rewriting process is used, and many different such processes can be considered. To better illustrate this, we consider a graph where each rewriting step corresponds to an edge, and thus a path corresponds to a rewriting process. In this paper, we consider the identification of paths, within the context of the membership problem, to propose a criterion of identification of proofs.
This article is part of the theme issue ‘The notion of ‘simple proof’ - Hilbert’s 24th problem’.

The algebraic variety defined by the idempotents of an incidence monoid is investigated. Its irreducible components are determined. The intersection with an antichain submonoid is shown to be the union of these irreducible components. The antichain monoids of bipartite posets are shown to be orthodox semigroups. The Green’s relations are explicitly determined, and applications to conjugacy problems are described. In particular, it is shown that two elements in the antichain monoid are primarily conjugate in the monoid if and only if they belong to the same -class and their multiplication by an idempotent of the same -class gives conjugate elements in the group.

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.

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.

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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.

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.

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.

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].

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.