有限维紧拓扑代数的分类，初步报告
Classification of finite-dimensional compact topological algebras, preliminary report
摘要：
拓扑空间$A$称为与方程组$Σ$相容（包含运算符号$F_t$）当且仅当存在连续运算$\overline F_t$在$A$上全同满足$Σ$。本文主要研究有限单纯复形$A$的相容关系。我们回顾和扩展已知的兼容性和不兼容性在这方面。许多这样的空间不与非平凡的$Σ$相容。本文最后提出了开放性问题。例如，如果$A$与$Σ$兼容，那么这一事实是否可以从$A$与某些只涉及三进制运算的$Γ$的兼容性推导出来？如果$A$与$Σ$兼容，那么操作$\overline F_t$可以被选择为分段多线性吗？相容关系（在有限$Σ$和有限复体$A$之间）是算法的吗？对于$A$，单形（即，一个封闭的实区间），是否有一些简单的特征集的所有$Σ$兼容$A$？
英文摘要：
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 $A$?
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
