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We consider sets of operations on a set A that are closed under permutation of variables, addition of dummy variables and composition. We describe these closed sets in terms of a Galois connection between operations and systems of pointed multisets, and we also describe the closed sets of the dual objects by means of necessary and sufficient closure conditions. Moreover, we show that the corresponding closure systems are uncountable for every A with at least two elements., 22 pages; Section 4 added

We establish the notion of a separating family of locale maps, which is the localic analogue of the topological concept of separating points from closed sets by continuous maps. We then present a localic version of the topological embedding (or diagonal) theorem. Applications to arbitrary locales, zero-dimensional locales, and completely regular locales are given. Using the axiom of choice, we are able to control the number of factors of the target localic products so that it does not exceed the weight of the embeddable locale. Apart from the proofs of results involving the weights of locales, the remaining proofs are valid in topos logic.

The paper deals with $\Sigma-$composition of terms, which allows us to extend the derivation rules in formal deduction of identities. The concept of essential variables and essential positions of terms with respect to a set of identities is a key step in the simplification of the process of formal deduction. $\Sigma-$composition of terms is defined as replacement between $\Sigma$-equal terms. This composition induces $\Sigma R-$deductively closed sets of identities. In analogy to balanced identities we introduce and investigate $\Sigma-$balanced identities for a given set of identities $\Sigma$., Comment: 17 pages, 2 figures

Pippenger's Galois theory of finite functions and relational constraints is extended to the infinite case. The functions involved are functions of several variables on a set $A$ and taking values in a possibly different set $B$, where any or both of $A$ and $B$ may be finite or infinite., Comment: arXiv admin note: text overlap with arXiv:0810.3212

A variety $\cal V $ of $ \Omega $ -algebras is said to be primitive if it is defined by a system of primitive identities, i.e. formulas of form (1.1). The main results are descriptions of closed set of primitive identities and of free objects in primitive varieties.

MARCEL WILD It will be shown that a complete join epimorphism F between two complete lattices L~ and L2 which maps principal ideals upon principal ideals preserves so-called meet weak lattice identities that hold in L1 (e.g. modularity, distributivity). Each onto map between two closure spaces (A, -) and (X, -) which is continuous and closed, induces a complete join epimorphism F of the above kind between the corresponding lattices of closed sets