Patrick J. Morandi

Profinite Completions and Canonical Extensions of Heyting Algebras

We show that the profinite completions and canonical extensions of bounded distributive lattices and of Boolean algebras coincide. We characterize dual spaces of canonical extensions of bounded distributive lattices and Heyting algebras in terms of Nachbin order-compactifications. We give the dual description of the profinite completion

Scattered, Hausdorff-reducible, and hereditarily irresolvable spaces

\widehat{H}

Maximal orders over valuation rings

of a Heyting algebra H, and characterize the dual space of

Noncrossed product division algebras with a Baer ordering

\widehat{H}

PROFINITE HEYTING ALGEBRAS AND PROFINITE COMPLETIONS OF HEYTING ALGEBRAS

. We also give a necessary and sufficient condition for the profinite completion of H to coincide with its canonical extension, and provide a new criterion for a variety V of Heyting algebras to be finitely generated by showing that V is finitely generated if and only if the profinite completion of every member of V coincides with its canonical extension. From this we obtain a new proof of a well-known theorem that every finitely generated variety of Heyting algebras is canonical.

The Priestley Separation Axiom for Scattered Spaces

Abstract We show that a topological space is hereditarily irresolvable if and only if it is Hausdorff-reducible. We construct a compact irreducible T 1 -space and a connected Hausdorff space, each of which is strongly irresolvable. Furthermore, we show that the three notions of scattered, Hausdorff-reducible, and hereditarily irresolvable coincide for a large class of spaces, including metric, locally compact Hausdorff, and spectral spaces.

Priestley Rings and Priestley Order-Compactifications

Abstract In this paper we study maximal orders over commutative valuation rings in central simple algebras. We are particularly interested in maximal orders which are either Bezout or semihereditary. We construct a class of Bezout maximal orders and a class of semihereditary maximal orders, and show that for any valuation ring V (resp. V with value group Z m), any Bezout (resp. semihereditary) maximal order over V belongs to the class constructed. Furthermore, we classify all maximal orders in M2(F) over a valuation ring with value group Z m and in Mn(F) given a mild “defectless” assumption.

Order-Compactifications of Totally Ordered Spaces: Revisited

Let n | m be positive integers with the same prime factors, such that p3 | n for some prime p. We construct a noncrossed product division algebra D with involution ∗, of index m and exponent n, such that D possesses a Baer ordering relative to the involution ∗. Using similar techniques we construct indecomposable division algebras with involution possessing a Baer

Ruler and Compass Constructions

Abstract This paper surveys recent developments in the theory of profinite Heyting algebras (resp. bounded distributive lattices, Boolean algebras) and profinite completions of Heyting algebras (resp. bounded distributive lattices, Boolean algebras). The new contributions include a necessary and sufficient condition for a profinite Heyting algebra (resp. bounded distributive lattice) to be isomorphic to the profinite completion of a Heyting algebra (resp. bounded distributive lattice). This results in simple examples of profinite bounded distributive lattices that are not isomorphic to the profinite completion of any bounded distributive lattice. We also show that each profinite Boolean algebra is isomorphic to the profinite completion of some Boolean algebra. It is still an open question whether each profinite Heyting algebra is isomorphic to the profinite completion of some Heyting algebra.

HYPERBOLICITY OF ALGEBRAS WITH INVOLUTION AND CONNECTIONS WITH CLIFFORD ALGEBRAS

Let R be a quasi-order on a compact Hausdorff topological space X. We prove that if X is scattered, then R satisfies the Priestley separation axiom if and only if R is closed in the product space X×X. Furthermore, if X is not scattered, then we show that there is a quasi-order on X that is closed in X×X but does not satisfy the Priestley separation axiom. As a result, we obtain a new characterization of scattered compact Hausdorff spaces.