Klaus Denecke

Galois Connections and Applications

Preface M. Erne Adjunctions and Galois Connections: Origins, History and Development G. Janelidze Categorical Galois Theory: Revision and Some Recent Developments M. Erne The Polarity between Approximation and Distribution K. Denecke, S.L. Wismath Galois Connections and Complete Sublattices R. Poschel Galois Connections for Operations and Relations K. Kaarli Galois Connections and Polynomial Completeness K. Glazek, St. Niwczyk Q-Independence and Weak Automorphisms A. Szendrei A Survey of Clones Closed Under Conjugation P. Burmeister Galois Connections for Partial Algebras K. Denecke, S.L. Wismath Complexity of Terms and the Galois Connection Id-Mod J. Lambek Iterated Galois Connections in Arithmetic and Linguistics I. Chajda, R. Halas Deductive Systems and Galois Connections J. Slapal A Galois Correspondence for Digital Topology W. Gahler Galois Connections in Category Theory, Topology and Logic R. Wille Dyadic Mathematics - Abstractions from Logical Thought Index

Complexity of terms, composition, and hypersubstitution

We consider four useful measures of the complexity of a term: the maximum depth (usually called the depth), the minimum depth, the variable count, and the operation count. For each of these, we produce a formula for the complexity of the composition Smn(s,t1,…,tn) in terms of the complexity of the inputs s, t1,…, tn. As a corollary, we also obtain formulas for the complexity of σˆ[t] in terms of the complexity of t when t is a compound term and σ is a hypersubstitution. We then apply these formulas to the theory of M-solid varieties, examining the k-normalization chains of a variety with respect to the four complexity measures.

REGULAR ELEMENTS AND GREEN'S RELATIONS IN MENGER ALGEBRAS OF TERMS

Defining an (n + 1)-ary superposition operation Sn on the set Wτ (Xn) of all n-ary terms of type τ , one obtains an algebra n − clone τ := (Wτ (Xn); Sn, x1, . . . , xn) of type (n + 1, 0, . . . , 0). The algebra n − clone τ is free in the variety of all Menger algebras ([9]). Using the operation Sn there are different possibilities to define binary associative operations on the set Wτ (Xn) and on the cartesian power Wτ (Xn) n. In this paper we study idempotent and regular elements as well as Green’s relations in semigroups of terms with these binary associative operations as fundamental operations.

Menger Algebras and Clones of Cooperations

The superposition of cooperations satisfies the well-known clone axioms (C1), (C2) and (C3). We define terms for indexed coalgebras of type τ, cooperations induced by those terms, and prove that the set of all induced cooperations forms a clone. This clone is equal to the clone of all cooperations generated by the fundamental cooperations of an indexed coalgebra. Finally, we introduce the concept of rational equivalence for coalgebras and determine all two-element coalgebras up to rational equivalence.

MULTI-HYPERSUBSTITUTIONS AND COLORED SOLID VARIETIES

Hypersubstitutions are mappings which map operation symbols to terms. Terms can be visualized by trees. Hypersubstitutions can be extended to mappings defined on sets of trees. The nodes of the trees, describing terms, are labelled by operation symbols and by colors, i.e. certain positive integers. We are interested in mappings which map differently-colored operation symbols to different terms. In this paper we extend the theory of hypersubstitutions and solid varieties to multi-hypersubstitutions and colored solid varieties. We develop the interconnections between such colored terms and multi-hypersubstitutions and the equational theory of Universal Algebra. The collection of all varieties of a given type forms a complete lattice which is very complex and difficult to study; multi-hypersubstitutions and colored solid varieties offer a new method to study complete sublattices of this lattice.

EQUATIONAL THEORIES GENERATED BY HYPERSUBSTITUTIONS OF TYPE (n)

Hypersubstitutions map n-ary operation symbols to n-ary terms. Such mappings can be uniquely extended to mappings defined on the set of all terms. It turns out that the kernels of hypersubstitution...

Separation of clones of cooperations by cohyperidentities

An n-ary cooperation is a mapping from a nonempty set A to the nth copower of A. A clone of cooperations is a set of cooperations which is closed under superposition and contains all injections. Coalgebras are pairs consisting of a set and a set of cooperations defined on this set. We define terms for coalgebras, coidentities and cohyperidentities. These concepts will be applied to give a new solution of the completeness problem for clones of cooperations defined on a two-element set and to separate clones of cooperations by coidentities.

Hyperformulas and solid algebraic systems

Defining a composition operation on sets of formulas one obtains a many-sorted algebra which satisfies the superassociative law and one more identity. This algebra is called the clone of formulas of the given type. The interpretations of formulas on an algebraic system of the same type form a many-sorted algebra with similar properties. The satisfaction of a formula by an algebraic system defines a Galois connection between classes of algebraic systems of the same type and collections of formulas. Hypersubstitutions are mappings sending pairs of operation symbols to pairs of terms of the corresponding arities and relation symbols to formulas of the same arities. Using hypersubstitutions we define hyperformulas. Satisfaction of a hyperformula by an algebraic system defines a second Galois connection between classes of algebraic systems of the same type and collections of formulas. A class of algebraic systems is said to be solid if every formula which is satisfied is also satisfied as a hyperformula. On the basis of these two Galois connections we construct a conjugate pair of additive closure operators and are able to characterize solid classes of algebraic systems.

HYPERSUBSTITUTIONS OF MANY-SORTED ALGEBRAS

Many-sorted algebras are used in Computer Science for abstract data type specifications. It is widely believed that many-sorted algebras are the appropriate mathematical tools to explain what abstract data types are ([1]). In this paper we want to extend the concept of a hypersubstitution from the one-sorted to the many-sorted case. Hypersubstitutions are used for one-sorted algebras to define hyperidentities and M-solid varieties ([2]). We will prove that extensions of hypersubstitutions for many-sorted algebras are endomorphisms of the many-sorted clone of a given type. As in the one-sorted case we define a binary operation for hypersubstitutions and prove that with respect to this operation all many-sorted hypersubstitutions form a monoid.

A Characterization of P-Compatible Varieties

P-Compatibility is a hereditary property of identities which generalizes the properties of normality and externality of identities. Chajda characterized the normalization of a variety by an algebraic construction called a choice algebra. In this paper, we generalize this characterization to the least P-compatible variety P(V) determined by a variety V for any partition P using P-choice algebras. We also study the clone of (strongly) P-compatible n-ary terms of a variety V, and relate identities of this clone to (strongly) P-compatible hyperidentities of the variety V.