Related papers: Coherence for plactic monoids via rewriting theory…

Polymorphism, the ability of a compound to crystallize in multiple distinct structures, plays a vital role in determining the physical, chemical, and functional properties of materials. Accurate identification and prediction of polymorphic…

Faces play a central role in the combinatorial and computational aspects of polyhedra. In this paper, we present the first formalization of faces of polyhedra in the proof assistant Coq. This builds on the formalization of a library…

Incorporating known symmetries in data into machine learning models has consistently improved predictive accuracy, robustness, and generalization. However, achieving exact invariance to specific symmetries typically requires designing…

We suggest new modification (we call it a noise reduction procedure) for Steinhardt parameters which are often used for detecting crystalline structures in computer simulation of solids and soft matter systems. We have also developed a new…

In this note we give a simple unifying proof of the undecidability of several diagrammatic properties of term rewriting systems that include: local confluence, strong confluence, diamond property, subcommutative property, and the existence…

Symmetric monoidal theories (SMTs) generalise algebraic theories in a way that make them suitable to express resource-sensitive systems, in which variables cannot be copied or discarded at will. In SMTs, traditional tree-like terms are…

This paper provides a combinatorial characterisation for generic forced symmetric rigidity of bar-joint frameworks in the Euclidean plane that are symmetric with respect to the orientation-reversing wallpaper group…

We give a purely combinatorial proof of the positivity of the stabilized forms of the generalized exponents associated to each classical root system. In finite type A_{n-1}, we rederive the description of the generalized exponents in terms…

We define and study an action of the symmetric group on the Yokonuma--Hecke algebra. This leads to the definition of two classes of algebras. The first one is connected with the image of the algebra of the braid group inside the…

In recent years, the realm of crystalline materials has witnessed a surge in the development of generative models, predominantly aimed at the inverse design of crystals with tailored physical properties. However, spatial symmetry, which…

In the context of varieties of representations of arbitrary quivers, possibly carrying loops, we define a generalization of Lusztig Lagrangian subvarieties. From the combinatorial study of their irreducible components arises a structure…

Monoids generated by elements of order two appear in numerous places in the literature. For example, Coxeter reflection groups in geometry, Kuratowski monoids in topology, various monoids generated by regular operations in language theory…

We give a method to prove confluence of term rewriting systems that contain non-terminating rewrite rules such as commutativity and associativity. Usually, confluence of term rewriting systems containing such rules is proved by treating…

We establish a formal correspondence between resource calculi an appropriate linear multicategories. We consider the cases of (symmetric) representable, symmetric closed and autonomous multicategories. For all these structures, we prove…

Using methods of homological algebra, we obtain an explicit crystal isomorphism between two realizations of crystal bases of the lower part of the quantized enveloping algebra of (almost all) finite dimensional simply-laced Lie algebras.…

This paper is addressed to logicians not familiar with category theory. It gives a new proof of coherence for symmetric monoidal closed categories, proven by Kelly and Mac Lane in early 1970s. We find this result of great importance for…

There are two parts to this work, which are largely independent. The first consists of a series of results concerning the crystal commutor of Henriques and Kamnitzer. We first describe the relationship between the crystal commutor and…

We introduce a computational method to optimize target physical properties in the full configuration space regarding atomic composition, chemical stoichiometry, and crystal structure. The approach combines the universal potential of the…

I motivate a variation (due to K. Szlach\'{a}nyi) of monoidal categories called skew-monoidal categories where the unital and associativity laws are not required to be isomorphisms, only natural transformations. Coherence has to be…

Recently the third named author defined a 2-parametric family of groups $G_n^k$ \cite{gnk}. Those groups may be regarded as a certain generalisation of braid groups. Study of the connection between the groups $G_n^k$ and dynamical systems…