Brian A. Davey

Duality Theory on Ten Dollars a Day

Duality theory grew out of two classical special cases—Pontryagin’s duality for abelian groups and Stone’s duality for Boolean algebras. In the late 1960s and early 1970s, it was further fertilized by Priestley’s duality for distributive lattices and the Hofmann-Mislove-Stralka duality for semilattices. Until the early 1970s, general approaches to duality theory were firmly rooted in category theory. The study of duality theory within general algebra began in the mid-1970s but blossomed in the 1980s. This paper presents an overview of duality theory from 1980 up to early 1992 as seen through the eyes of an algebraist. The presentation is in the style of a travel guide and is aimed at beginning graduate students. A minimum of general algebra and topology is assumed and category theory is completely avoided.

Sheaf spaces and sheaves of universal algebras

In recent years it has become popular to use sheaf spa :es to obtain representation theorems and/or embedding theorems for various alge- bras-for example: semigroups (Adams [1], Keimel [15, 17]), rings (Davis [9], Dauns and Hofmann [6], Hofmann [12], Kist [21], Mew- born [22], Mulvey [24], Peercy [25], Pierce [26]),/-groups (Davis [8], Keimel [16, 18]), f-rings (Keimel [13, 14, 16, 18]), distributive lattices (Davey [7]), and for universal algebras (Comer [3, 4], Keimel [15], Keimel and Werner [19]). See also the extensive list of references in [12]. Thus it is natural to investigate to what extent the various constructions used concur. We carry out our investigation in four parts. In Section 1 we present the basic properties of sheaf spaces and sheaves of universal algebras. In Section 2 we give a general procedure for the conversion of a subdirect product representation of an algebra into a representation as an algebra of global sections of a sheaf space; this generalizes the procedure used in all of the papers mentioned above. Section 3 gives a brief study of the elementary properties of sheaf spaces and sheaves over Boolean spaces, and in Section 4 we apply our results to construct and study the extension FA of an algebra A induced by a Boolean algebra of congruences. The constructions of [1, 7, 8, 9, 14, 17, 21], and [25] then arise as particular cases. Under certain conditions the extension is in fact an isomorphism, in which case the general representation theorem of Comer [3] is obtained. Finally, we prove that F A has a (choice free) construction as a direct limit, showing that each of the constructions listed above may be obtained in a manner analogous to that for/-groups in Conrad [5] and for semiprime rings in Speed [28]. We use standard universal algebra terminology. The lattice of congruences on an algebra A is denoted by I

Exponentiation and Duality

(A), with least element co and greatest element ~. If

The Syntax and Semantics of Entailment in Duality Theory

In a unified treatment of cardinal and ordinal arithmetic G. Birkhoff [18], [20] defined (cardinal) exponentiation of ordered sets: for ordered sets X and Y, X Y (the power) is the set of all order-preserving maps of Y (the exponent) to X (the base) ordered componentwise. Our aim. is to review a significant body of results concerning powers of ordered sets and to present some central open problems arising in recent work. Roughly, we have two topics: first, an analysis of the structure of powers and their symmetries; second, a study of duality results for lattice-ordered algebras.

Dualizability and graph algebras

Both syntactic and semantic solutions are given for the entailment problem of duality theory. The test algebra theorem provides both a syntactic solution to the entailment problem in terms of primitive positive formulae and a new derivation of the corresponding result in clone theory, viz. the syntactic description of Inv(Pol(R)) for a given set R of finitary relations on a finite set. The semantic solution to the entailment problem follows from the syntactic one, or can be given in the form of an algorithm. It shows, in the special case of a purely relational type, that duality-theoretic entailment is describable in terms of five constructs, namely trivial relations, intersection, repetition removal, product, and refractive projection. All except the last are concrete, in the sense that they are described by a quantifier-free formula. It is proved that if the finite algebra M generates a congruence-distributive variety and all subalgebras of M are subdirectly irreducible, then concrete constructs suffice to describe entailment. The concept of entailment appropriate to strong dualities is also introduced, and described in terms of coordinate projections, restriction of domains, and composition of partial functions. ?

Monotone clones and congruence modularity

Abstract We characterize the finite graph algebras which are dualizable. Indeed, a finite graph algebra is dualizable if and only if each connected component of the underlying graph is either complete or bipartite complete (or a single point).

A characterization of semi-distributivity

We investigate the relationship between the local shape of an ordered set P=(P; ≤) and the congruence-modularity of the variety V generated by an algebra A=(P; F) each of whose operations is order-preserving with respect to P. For example, if V is k-permutable (k≥2) then P is an antichain; if P is both up and down directed and V is congruence-modular, then V is congruence-distributive; if A is a dual discriminator algebra, then either P is an antichain or a two-element chain. We also give a useful necessary condition on P for V to be congruence-modular. Finally a class of ordered sets called braids is introduced and it is shown that if P is a braid of length 1, in particular if P is a crown, then the variety V is not congruence-modular.

Endoprimal distributive lattices are endodualisable

It is easy to verify that the lattices L 1, L2, L 3 and L4, illustrated in Figure 1, and the duals L3 ~ and L~, are not semi-distributive. We show that these lattices are characteristic of the failure of semi-distributivity. THEOREM. A lattice L of finite length is semi-distributive if and only if it contains no sublattice isomorphic to LI, L2, L3, L~, L4 or L~; in fact, L satisfies (SDv) / f and only if it has no sublattice isomorphic to L 1, L2, La, L~ or L4. It is easily verified that the lattice illustrated in Figure 3 (cf. R. Wille I-5]) satisfies neither (SDv) nor (SD^) but has no sublattice isomorphic to L1, L2, L3, L~, L4 or L~. Hence, in general, the Theorem fails for lattices with infinite chains.

The quest for strong dualities

AbstractL. Márki and R. Pöschel have characterised the endoprimal distributive lattices as those which are not relatively complemented. The theory of natural dualities implies that any finite algebraA on which the endomorphisms of A yield a duality on the quasivariety