We consider three monads on $\mathsf{Top}$ , the category of topological spaces, which formalize topological aspects of probability and possibility in categorical terms. The first one is the Hoare hyperspace monad H, which assigns to every space its space of closed subsets equipped with the lower Vietoris topology. The second one is the monad V of continuous valuations, also known as the extended probabilistic powerdomain. We construct both monads in a unified way in terms of double dualization. This reveals a close analogy between them and allows us to prove that the operation of taking the support of a continuous valuation is a morphism of monads $V \to H$ . In particular, this implies that every H-algebra (topological complete semilattice) is also a V-algebra. We show that V can be restricted to a submonad of $\tau$ -smooth probability measures on $\mathsf{Top}$ . By composing these morphisms of monads, we obtain that taking the supports of $\tau$ -smooth probability measures is also a morphism of monads.

Ordering is a well-established concept in mathematics and also plays an important role in many areas of computer science, where quasi-orderings, most notably well-founded quasi-orderings and well-quasi-orderings, are of particular interest. This paper deals with quasi-orderings on first-order terms and introduces a new notion of unification based on a special quasi-order, known as homeomorphic tree embedding. Historically, the development of unification theory began with the central notion of a most general unifier based on the subsumption order. A unifier $\sigma$ is most general, if it subsumes any other unifier $\tau$ , that is, if there is a substitution $\lambda$ with $\tau=_{E}\sigma\lambda$ , where E is an equational theory and $=_{E}$ denotes equality under E. Since there is in general more than one most general unifier for unification problems under equational theories E, called E-Unification, we have the notion of a complete and minimal set of unifiers under E for a unification problem $\varGamma$ , denoted as $\mu\mathcal{U}\Sigma_{E}(\Gamma)$ . This set is still the basic notion in unification theory today. But, unfortunately, the subsumption quasi-order is not a well-founded quasi-order, which is the reason why for certain equational theories there are solvable E-unification problems, but the set $\mu\mathcal{U}\Sigma_{E}(\Gamma)$ does not exist. They are called type nullary in the unification hierarchy. In order to overcome this problem and also to substantially reduce the number of most general unifiers, we extended the well-known encompassment order on terms to an encompassment order on substitutions (modulo E). Unification under the encompassment order is called essential unification and if $\mu\mathcal{U}\Sigma_{E}(\Gamma)$ exists, then the complete set of essential unifiers $e\mathcal{U}\Sigma_{E}(\Gamma)$ is a subset of $\mu\mathcal{U}\Sigma_{E}(\Gamma)$ . An interesting effect is that many E-unification problems with an infinite set of most general unifiers (under the subsumption order) reduce to a problem with only finitely many essential unifiers. Moreover, there are cases of an equational theory E, for which the complete set of most general unifiers does not exist, the minimal and complete set of essential unifiers however does exist. Unfortunately again, the encompassment order is not a well-founded quasi-ordering either, that is, there are still theories with a solvable unification problem, for which a minimal and complete set of essential unifiers does not exist. This paper deals with a third approach, namely the extension of the well-known homeomorphic embedding of terms to a homeomorphic embedding of substitutions (modulo E). We examine the set of most general, minimal, and complete E-unifiers under the quasi-order of homeomorphic embedment modulo an equational theory E, called $\varphi U\Sigma_{E}(\Gamma)$ , and propose an appropriate definitional framework based on the standard notions of unification theory extended by notions for the tree embedding theorem or Kruskal’s theorem as it is called. The main results are that for regular theories the minimal and complete set $\varphi\mathcal{U}\Sigma_{E}(\Gamma)$ always exists. If we restrict the E-embedding order to pure E-embedding, a well-known technique in logic programming and term rewriting where the difference between variables is ignored, the set $\varphi_{\pi}\mathcal{U}\Sigma_{E}(\Gamma)$ always exists and it is even finite for any theory E.

We show how concurrent quantales and concurrent Kleene algebras arise as convolution algebras of functions from relational structures with two ternary relations that satisfy relational interchange laws into concurrent quantales or Kleene algebras, among others. The elements of the quantales can be understood as weights; the case where weights are drawn from the booleans corresponds to languages. We develop a correspondence theory between properties of the relational structures and algebraic properties in the weight and convolution algebras in the sense of modal and substructural logics, or boolean algebras with operators. The resulting correspondence triangles yield in particular general construction principles for models of concurrent quantales and Kleene algebras as convolution algebras from much simpler relational structures, including weighted ones for quantitative applications. As examples, we construct the concurrent quantales and Kleene algebras of weighted words, digraphs, posets, isomorphism classes of finite digraphs and pomsets.