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We call a simplicial complex algebraically rigid if its Stanley-Reisner ring admits no nontrivial infinitesimal deformations, and call it inseparable if does not allow any deformation to other simplicial complexes. Algebraically rigid simplicial complexes are inseparable. In this paper we study inseparability and rigidity of Stanley-Reisner rings, and apply the general theory to letterplace ideals as well as to edge ideals of graphs. Classes of algebraically rigid simplicial complexes and graphs are identified.
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The goal of the present paper is the study of some algebraic invariants of Stanley-Reisner rings of Cohen-Macaulay simplicial complexes of dimension $d - 1$. We prove that the inequality $d leq mathrm{reg}(Delta) cdot mathrm{type}(Delta)$ holds for any $(d-1)$-dimensional Cohen-Macaulay simplicial complex $Delta$ satisfying $Delta=mathrm{core}(Delta)$, where $mathrm{reg}(Delta)$ (resp. $mathrm{type}(Delta)$) denotes the Castelnuovo-Mumford regularity (resp. Cohen-Macaulay type) of the Stanley-Reisner ring $Bbbk[Delta]$. Moreover, for any given integers $d,r,t$ satisfying $r,t geq 2$ and $r leq d leq rt$, we construct a Cohen-Macaulay simplicial complex $Delta(G)$ as an independent complex of a graph $G$ such that $dim(Delta(G))=d-1$, $mathrm{reg}(Delta(G))=r$ and $mathrm{type}(Delta(G))=t$.
Let $Delta$ be a one-dimensional simplicial complex. Let $I_Delta$ be the Stanley-Reisner ideal of $Delta$. We prove that for all $s ge 1$ and all intermediate ideals $J$ generated by $I_Delta^s$ and some minimal generators of $I_Delta^{(s)}$, we have $${rm reg} J = {rm reg} I_Delta^s = {rm reg} I_Delta^{(s)}.$$
Simplicial complexes are a versatile and convenient paradigm on which to build all the tools and techniques of the logic of knowledge, on the assumption that initial epistemic models can be described in a distributed fashion. Thus, we can define: knowledge, belief, bisimulation, the group notions of mutual, distributed and common knowledge, and also dynamics in the shape of simplicial action models. We give a survey on how to interpret all such notions on simplicial complexes, building upon the foundations laid in prior work by Goubault and others.
The paper studies the connectivity properties of facet graphs of simplicial complexes of combinatorial interest. In particular, it is shown that the facet graphs of $d$-cycles, $d$-hypertrees and $d$-hypercuts are, respectively, $(d+1)$, $d$, and $(n-d-1)$-vertex-connected. It is also shown that the facet graph of a $d$-cycle cannot be split into more than $s$ connected components by removing at most $s$ vertices. In addition, the paper discusses various related issues, as well as an extension to cell-complexes.
In the spirit of topological entropy we introduce new complexity functions for general dynamical systems (namely groups and semigroups acting on closed manifolds) but with an emphasis on the dynamics induced on simplicial complexes. For expansive systems remarkable properties are observed. Known examples are revisited and new examples are presented.
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