arXiv subjects
Publications and source records attributed to Walter Taylor.
For a metric space $(A,d)$, and a set $Σ$ of equations, some quantities are introduced that measure the size of discontinuities that must occur in operations satisfying $Σ$ (identically) on $A$. We are able to evaluate these quantities in a few easy cases.}
For a metric space $(A,d)$, and a set $Σ$ of equations, a quantity is introduced that measures how far continuous operations must deviate from satisfying $Σ$ on $(A,d)$. }
A topological space $A$ is said to be compatible with a set $Σ$ of equations (involving operation symbols $F_t$) iff there are continuous operations $\overline F_t$ identically satisfying $Σ$ on $A$. The paper's main focus is on the compatibility relation for $A$ a finite simplicial complex. We review and extend the known compatibilities and incompatibilities in this context. Many such spaces are compatible with no non-trivial $Σ$. The paper concludes with open questions. For example, if $A$ is compatible with $Σ$, can this fact be deduced from the compatibility of $A$ with some $Γ$ that involves only ternary operations? If $A$ is compatible with $Σ$, can the operations $\overline F_t$ be chosen as piecewise multilinear? Is the compatibility relation (between finite $Σ$ and finite complexes $A$) algorithmic? For $A$ a one-simplex (i.e., a closed real interval), is there some simple characterization of the set of all $Σ$ compatible with $A$?