Finite posets and topological spaces in locally finite varieties (Q2577685)

It is proved that if a finite connected poset admits an order-preserving Taylor operation, then it has all homotopy groups trivial. This is deduced from the result of \textit{W. Taylor} [Can. J. Math. 29, 498--527 (1977; Zbl 0357.08004)] saying that such a poset has an abelian fundamental group. As a consequence, the authors deduce omitting-types theorems for locally finite varieties \(\mathcal V\) in terms of posets carrying an algebra in \(\mathcal V\).