Algebras and Coalgebras, Edition: version 19 Oct 2005 by Yde Venema

By Yde Venema

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Example text

12 As suggested by notation, the topology σ is closely connected to the kind of inclusion of B in Bσ . Let us just mention a couple of salient facts here. First, it is easy to see that the set {[a, b] | a, b ∈ B} is a basis for σ. This reveals that the set B is topologically dense in σ, in the sense that every σ-open set contains an element of B. But also, B constitutes the collection of isolated points of σ — recall that a point x is isolated in a topology if the singleton {x} is open. It is the latter two properties that make it possible to extend arbitrary maps between Boolean algebras to their extensions; we will come back to this at the end of this section.

Sets containing no pair of neighboring vertices. The chromatic number χ(G) of G is the smallest number k for which it has a k-coloring, and ∞ if it has no finite coloring. A cycle in G is a path x1 Ex2 E . . Exn Ex1 such that n ≥ 3 and x1 , . . , xn are all distinct vertices; the length of this cycle is n. Now intuitively, a lack of short cycles, indicating a certain ‘looseness’ of the graph, should make it easy ˝ [22] reveals the existence of a sequence of finite graphs whose to color a graph with few colors, but E RD OS n-th member Gn has chromaticity bigger than n while Gn has no cycles of length ≤ n.

Algebraically, these applications are connected to the following result. 12 Let V be a variety of Boolean algebras with τ -operators. If V is canonical, then V is complete. Proof. If V is canonical then V ⊆ SCmCst(V) so clearly V is generated by its perfect members. ✷ So where do we find canonical varieties? In general there seem to be two roads here, a syntactic and a model-theoretic one. The syntactic approach is the most important one for applications. Basically, the idea is to find out whether a logic is canonical on the basis of the syntactic shape of the axioms.

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