@article {doi:10.1080/00927872.2021.1955901, title = {Identities and bases in the hypoplactic monoid}, journal = {Communications in Algebra}, volume = {50}, number = {1}, year = {2022}, note = {n/a}, pages = {146-162}, publisher = {Taylor \& Francis}, abstract = {n/a}, doi = {10.1080/00927872.2021.1955901}, url = {https://doi.org/10.1080/00927872.2021.1955901}, author = {Cain, Alan J. and Malheiro, Ant{\'o}nio and Duarte Ribeiro} } @article {12846, title = {Identification of proofs via syzygies}, journal = {Philosophical Transactions of the Royal Society A}, volume = {377}, year = {2019}, abstract = {

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 Gr{\"o}bner 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 {\textquoteleft}The notion of {\textquoteleft}simple proof{\textquoteright} - Hilbert{\textquoteright}s 24th problem{\textquoteright}.

}, url = {http://dx.doi.org/10.1098/rsta.2018.0275}, author = {Malheiro, Ant{\'o}nio and Jos{\'e} Francisco Reis} } @inbook {Cain2017, title = {Combinatorics of Cyclic Shifts in Plactic, Hypoplactic, Sylvester, and Related Monoids}, booktitle = {Combinatorics on Words: 11th International Conference, WORDS 2017, Montr{\'e}al, QC, Canada, September 11-15, 2017, Proceedings}, year = {2017}, note = {

n/a

}, pages = {190{\textendash}202}, publisher = {Springer International Publishing}, organization = {Springer International Publishing}, address = {Cham}, 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.

}, isbn = {978-3-319-66396-8}, doi = {10.1007/978-3-319-66396-8_18}, url = {https://doi.org/10.1007/978-3-319-66396-8_18}, author = {Cain, Alan J. and Malheiro, Ant{\'o}nio}, editor = {Brlek, Sre{\v c}ko and Dolce, Francesco and Reutenauer, Christophe and Vandomme, {\'E}lise} }