Finite posets and topological spaces in locally finite varieties

We prove that if a finite connected poset admits an order-preserving Taylor operation, then all of its homotopy groups are trivial. We use this to give new characterisations of locally finite varieties omitting type 1 in terms of the posets (or equivalently, finite topological spaces) in the variety. Similar variants of other omitting-type theorems are presented. We give several examples of posets that admit various types of Taylor operations; in particular, we exhibit a topological space which is not an H-space but is compatible with a set of non-trivial identities, answering a question of W. Taylor.