John Harding

New Mexico State University | NMSU · Department of Mathematical Sciences

An irreducible complete atomic oml of infinite height cannot be algebraic and have the covering property. However, modest departure from these conditions allows infinite-height examples. We use an extension of Kalmbach’s construction to the setting of infinite chains to provide an example of an infinite-height, irreducible, algebraic oml with the 2...

Quantum cylindric algebras were introduced by the author as a generalization of their classical counterpart. Primary examples based on Weaver’s treatment of the tensor power of a Hilbert space with diagonals given by symmetric tensor products are called full quantum cylindric set algebras. We generalize a result of Sudkamp in the classical setting...

An irreducible complete atomic OML of infinite height cannot both be algebraic and have the covering property. However, Kalmbach's construction provides an example of such an OML that is algebraic and has the 2-covering property, and Keller's construction provides an example of such an OML that has the covering property and is completely hereditari...

We introduce the category of stable compactications of T 0-spaces and obtain a dual description of it in terms of what we call Raney extensions of proximity frames. These are proximity frame embeddings of a regular proximity frame into a Raney lattice , i.e. the lattice of upsets of a poset. This duality generalizes the duality between compacticati...

The Vietoris space of a Stone space plays an important role in the coalgebraic approach to modal logic. When generalizing this to positive modal logic, there is a variety of relevant hyperspace constructions based on various topologies on a Priestley space and mechanisms to topologize the hyperspace of closed sets. A number of authors considered hy...

We introduce quantum monadic and quantum cylindric algebras. These are adaptations to the quantum setting of the monadic algebras of Halmos, and cylindric algebras of Henkin, Monk and Tarski, that are used in algebraic treatments of classical and intuitionistic predicate logic. Primary examples in the quantum setting come from von Neumann algebras...

The Priestley space X = (X, \pi , \le) of a bounded distributive lattice L carries three natural topologies, the Stone topology {\pi}, the topology of open upsets, and the topology of open downsets. For each of these topologies, there are three hyperspace topologies on its closed sets, the hit, miss, and hit-or-miss topologies. We give a systematic...

There is current interest in using ideas from quantum mechanics in the study of economics. We give an overview of an approach to quantum mechanics rooted not necessarily in Hilbert space, but in the primitive mathematical idea of direct products. This approach includes the standard von Neumann Hilbert space approach. It provides a conceptually simp...

In view of current interests in borrowing quantum probability notion in quantum mechanics to model uncertainty in economic activities and models, consistent with behavioral economics, we address in this paper an important, but missing, ingredient in quantum probability logic for reasoning and inference, namely quantum implication operator as a bona...

We survey several problems related to logical aspects of quantum structures. In particular, we consider problems related to completions, decidability and axiomatizability, and embedding problems. The historical development is described, as well as recent progress and some suggested paths forward.

A pseudo ordered set (X,≤) is a set X with a binary relation ≤ that is reflexive and antisymmetric. We associate to a pseudo ordered set X, a partially ordered set Γ(X) called the covering poset. Taking any completion (C,f) of the covering poset Γ(X), and a special equivalence relation 𝜃 on this completion, yields a completion C/𝜃 of the pseudo ord...

We introduce quantum monadic and quantum cylindric algebras. These are adaptations to the quantum setting of the monadic algebras of Halmos, and cylindric algebras of Henkin, Monk and Tarski, that are used in algebraic treatments of classical and intuitionistic predicate logic. Primary examples in the quantum setting come from von Neumann algebras...

In the context of quantum probability models for causal reasoning in economics, we discuss the mathematics surrounding classical conditional probability and its quantum counterpart for commuting events as given by Lüders rule. While there are many issues in terms of meaning and interpretation in both the classical and quantum setting, there is a tr...

We provide a simple proof of a recent result of Morton and van Alten that the canonical extension of a bounded distributive lattice is its free completely distributive extension. We show that this can be used to easily obtain a number of results, both known and new.

We survey several problems related to logical aspects of quantum structures. In particular, we consider problems related to completions, decidability and axiomatizability, and embedding problems. The historical development is described, as well as recent progress and some suggested paths forward.

Topos quantum mechanics, developed by Döring (2008); Döring and Harding Houston J. Math. 42(2), 559–568 (2016); Döring and Isham (2008); Flori 2013)); Flori (2018); Isham and Butterfield J. Theoret. Phys. 37, 2669–2733 (1998); Isham and Butterfield J. Theoret. Phys. 38, 827–859 (1999); Isham et al. J. Theoret. Phys. 39, 1413–1436 (2000); Isham and...

A Heyting algebra is supplemented if each element a has a dual pseudo-complement \(a^+\), and a Heyting algebra is centrally supplement if it is supplemented and each supplement is central. We show that each Heyting algebra has a centrally supplemented extension in the same variety of Heyting algebras as the original. We use this tool to investigat...

A poset P forms a locally compact \(T_0\)-space in its Alexandroff topology. We consider the hit-or-miss topology on the closed sets of P and the associated Fell compactification of P. We show that the closed sets of P with the hit-or-miss topology is the Priestley space of the bounded distributive lattice freely generated by the order dual of P. T...

In this note we adapt the treatment of topological spaces via Kuratowski closure and interior operators on powersets to the setting of \(T_0\)-spaces. A Raney lattice is a complete completely distributive lattice that is generated by its completely join prime elements. A Raney algebra is a Raney lattice with an interior operator whose fixpoints com...

It is known that fuzzy set theory can be viewed as taking place within a topos. There are several equivalent ways to construct this topos, one is as the topos of étalé spaces over the topological space \(Y=[0,1)\) with lower topology. In this topos, the fuzzy subsets of a set X are the subobjects of the constant étalé \(X\times Y\) where X has the...

A Heyting algebra is supplemented if each element $a$ has a dual pseudo-complement $a^+$, and a Heyting algebra is centrally supplement if it is supplemented and each supplement is central. We show that each Heyting algebra has a centrally supplemented extension in the same variety of Heyting algebras as the original. We use this tool to investigat...

We consider an alternate form of the equivalence between the category of compact Hausdorff spaces and continuous functions and a category formed from Gleason spaces and certain relations. This equivalence arises from the study of the projective cover of a compact Hausdorff space. This line leads us to consider the category of compact Hausdorff spac...

We develop a direct method to recover an orthoalgebra from its poset of Boolean subalgebras. For this a new notion of direction is introduced. Directions are also used to characterize in purely order-theoretic terms those posets that are isomorphic to the poset of Boolean subalgebras of an orthoalgebra. These posets are characterized by simple cond...

Based on results of Harding, Heunen, Lindenhovius and Navara, (2019), we give a connection between the category of AW*-algebras and their normal Jordan homomorphisms and a category COG of orthogemetries, which are structures that are somewhat similar to projective geometries, consisting of a set of points and a set of lines, where each line contain...

Topos quantum mechanics, developed by Isham et. al., creates a topos of presheaves over the poset V(N) of abelian von Neumann subalgebras of the von Neumann algebra N of bounded operators associated to a physical system, and established several results, including: (a) a connection between the Kochen-Specker theorem and the non-existence of a global...

In 1998 the author showed that the canonical extension of a bounded modular lattice need not be modular. The proof was indirect, using a deep result of Kaplansky. In this note we give an explicit example.

It is known that fuzzy set theory can be viewed as taking place within a topos. There are several equivalent ways to construct this topos, one is as the topos of \'{e}tal\'{e} spaces over the topological space $Y=[0,1)$ with lower topology. In this topos, the fuzzy subsets of a set $X$ are the subobjects of the constant \'{e}tal\'{e} $X\times Y$ wh...

A lattice P is transferable for a class of lattices K if whenever P can be embedded into the ideal lattice IK of some K∈ K, then P can be embedded into K. There is a rich theory of transferability for lattices. Here we introduce the analogous notion of MacNeille transferability, replacing the ideal lattice IK with the MacNeille completion K¯. Basic...

For a complete lattice $L$ and a relational structure $\mathfrak{X}=(X,(R_i)_I)$, we introduce the convolution algebra $L^{\mathfrak{X}}$. This algebra consists of the lattice $L^X$ equipped with an additional $n_i$-ary operation $f_i$ for each $n_i+1$-ary relation $R_i$ of $\mathfrak{X}$. For $\alpha_1,\ldots,\alpha_{n_i}\in L^X$ and $x\in X$ we s...

In Harding (Trans. Amer. Math. Soc. 348(5), 1839–1862 1996) it was shown that the direct product decompositions of any non-empty set, group, vector space, and topological space X form an orthomodular poset Fact X. This is the basis for a line of study in foundational quantum mechanics replacing Hilbert spaces with other types of structures. Here we...

A Birkhoff system is an algebra that has two binary operations ⋅ and + , with each being commutative, associative, and idempotent, and together satisfying x⋅(x + y) = x+(x⋅y). Examples of Birkhoff systems include lattices, and quasilattices, with the latter being the regularization of the variety of lattices. A number of papers have explored the bo...

This is the second part of a two-part paper on Birkhoff systems. A Birkhoff system is an algebra that has two binary operations ⋅ and + , with each being commutative, associative, and idempotent, and together satisfying x⋅(x + y) = x+(x⋅y). The first part of this paper described the lattice of subvarieties of Birkhoff systems. This second part cont...

For a complete lattice $L$ and a relational structure $\mathfrak{X}=(X,(R_i)_I)$, we introduce the convolution algebra $L^{\mathfrak{X}}$. This algebra consists of the lattice $L^X$ equipped with an additional $n_i$-ary operation $f_i$ for each $n_i+1$-ary relation $R_i$ of $\mathfrak{X}$. For $\alpha_1,\ldots,\alpha_{n_i}\in L^X$ and $x\in X$ we s...

We prove that the topology of a compact Hausdorff topological Heyting algebra is a Stone topology. It then follows from known results that a Heyting algebra is profinite iff it admits a compact Hausdorff topology that makes it a compact Hausdorff topological Heyting algebra.

It is well known that the closed subspaces of a Hilbert space form an orthomodular lattice. Elements of this orthomodular lattice are the propositions of a quantum mechanical system represented by the Hilbert space, and by Gleason’s theorem atoms of this orthomodular lattice correspond to pure states of the system. Wigner’s theorem says that the au...

This monograph treats the foundations of type-2 fuzzy sets. The mathematics involved is varied and interesting, with topics from lattice theory, universal algebra, logic, and category theory. However no previous background is assumed past undergraduate calculus and matrices. Notable byproducts are the decidability of equations for type-2 fuzzy sets...

For von Neumann algebras M;N not isomorphic to C C and without type I2 summands, we show that for an order-isomorphism f: AbSub M → AbSub N between the posets of abelian von Neumann subalgebras of M and N, there is a unique Jordan ∗-isomorphism g:M → N with the image g[S] equal to f(S) for each abelian von Neumann subalgebra S of M. The converse al...

The main concern of this paper is with the equations satisfied by the algebra of truth values of type-2 fuzzy sets. That algebra has elements all mappings from the unit interval into itself with operations given by certain convolutions of operations on the unit interval. There are a number of positive results. Among them is a decision procedure, si...

Since the work of Crown (J. Natur. Sci. Math. 15(1–2), 11–25 1975) in the 1970’s, it has been known that the projections of a finite-dimensional vector bundle E form an orthomodular poset (omp) P(E)$\mathcal {P}(E)$. This result lies in the intersection of a number of current topics, including the categorical quantum mechanics of Abramsky and Coeck...

Papert Strauss (Proc. London Math. Soc. 18(3), 217–230, 1968) used Pontryagin duality to prove that a compact Hausdorff topological Boolean algebra is a powerset algebra. We give a more elementary proof of this result that relies on a version of Bogolyubov’s lemma.

The elements of the truth value algebra of type-2 fuzzy sets are the mappings of the unit interval into itself, with operations given by various convolutions of the pointwise operations. This algebra can be specialized and generalized in various interesting ways. First, we consider the more general case of all mappings of a bounded chain with an in...

We define a 2-category whose objects are fuzzy sets and whose maps are relations subject to certain natural conditions. We enrich this category with additional monoidal and involutive structure coming from t-norms and negations on the unit interval. We develop the basic properties of this category and consider its relation to other familiar categor...

It is well known that the category KHaus of compact Hausdorff spaces is dually equivalent to the category KRFrm of compact regular frames. By de Vries duality, KHaus is also dually equivalent to the category DeV of de Vries algebras, and so DeV is equivalent to KRFrm, where the latter equivalence can be described constructively through Booleanizati...

In a classic paper, Smirnov [14] characterized the poset of compactifications of a completely regular space in terms of the proximities on the space. Later, Smyth [15] introduced the notion of a stable compactification of a T 0-space and described them in terms of quasi-proximities on the space. Banaschewski [1] formulated Smirnov's results in the...

There is a family of constructions to produce orthomodular structures from modular lattices, lattices that are M and M*-symmetric, relation algebras, the idempotents of a ring, the direct product decompositions of a set or group or topological space, and from the binary direct product decompositions of an object in a suitable type of category. We s...

Harding showed that the direct product decompositions of many different types of structures, such as sets, groups, vector spaces, topological spaces, and relational structures, naturally form orthomodular posets. When applied to the direct product decompositions of a Hilbert space, this construction yields the familiar orthomodular lattice of close...

For $\Bbb {F}$ the field of real or complex numbers, let $CG(\Bbb {F})$ be the continuous geometry constructed by von Neumann as a limit of finite dimensional projective geometries over $\Bbb {F}$ . Our purpose here is to show the equational theory of $CG(\Bbb {F})$ is decidable.

We discuss issues related to constructing an orthomodular structure from an object in a category. In particular, we consider axiomatics related to Baer *-semigroups, partial semigroups, and various constructions involving dagger categories, kernels, and biproducts.

In Bezhanishvili et al. (2012) we introduced the category MKHaus of modal compact Hausdorff spaces, and showed these were concrete realizations of coalgebras for the Vietoris functor on compact Hausdorff spaces, much as modal spaces are coalgebras for the Vietoris functor on Stone spaces. Also in Bezhanishvili et al. (2012) we introduced the catego...

The elements of the truth value algebra of type-2 fuzzy sets are all mappings of the unit interval into itself, with operations given by various convolutions of the pointwise operations. This algebra can be specialized and generalized in various interesting ways. Here we replace each copy of the unit interval by a finite chain, and define operation...

Wilce introduced the notion of a topological orthomodular poset and proved any compact topological orthomodular poset whose underlying orthomodular poset is a Boolean algebra is a topological Boolean algebra in the usual sense. Wilce asked whether the compactness assumption was necessary for this result. We provide an example to show the compactnes...

We study two categories, both having interval-valued fuzzy sets as objects. One has certain functions between the domains as morphisms and the other is expanded to include certain relations between the domains as morphisms. We describe some of the basic properties of each of these categories. We lift t-norms and negations to the category with relat...

An algebra with two binary operations · and + that are commutative, associative, and idempotent is called a bisemilattice. A bisemilattice that satisfies Birkhoff’s equation x · (x + y) = x + (x · y) is a Birkhoff system. Each bisemilattice determines, and is determined by, two semilattices, one for the operation + and one for the operation ·. A bi...

We introduce modal compact Hausdorff spaces as generalizations of modal spaces, and show these are coalgebras for the Vietoris functor on compact Hausdorff spaces. Modal compact regular frames and modal de Vries algebras are introduced as algebraic counterparts of modal compact Hausdorff spaces, and dualities are given for the categories involved....

A variety of lattices admits a meet dense regular completion if every lattice in the variety can be embedded into a complete lattice in the variety by an embedding that is meet dense and regular (preserves existing joins and meets). We show that exactly two varieties of lattices admit a meet dense regular completion, the variety of one-element latt...

Interpreting modal diamond as the closure of a topological space, we axiomatize the modal logic of each metrizable Stone space and of each extremally disconnected Stone space. As a corollary, we obtain that S4.1 is the modal logic of the Pelczynski compactification of the natural numbers and S4.2 is the modal logic of the Gleason cover of the Canto...

This paper is a continuation of the study of the variety generated by the truth value algebra of type-2 fuzzy sets. That variety and some of its reducts were shown to be generated by finite algebras, and in particular to be locally finite. A basic question remaining is whether or not these algebras have finite equational bases, and that is our prin...

For von Neumann algebras $\mathcal{M},\mathcal{N}$ without type $I_2$ summands, we show that for an order-isomorphism $f:AbSub \mathcal{M}\to AbSub \mathcal{N}$ between the posets of abelian von Neumann subalgebras of $\mathcal{M}$ and $\mathcal{N}$, there is a unique Jordan *-isomorphism $g:\mathcal{M}\to \mathcal{N}$ with the image $g[\mathcal{S}...

Sachs showed that a Boolean algebra is determined by its lattice of subalgebras. We establish the corresponding result for orthomodular lattices. We show that an orthomodular lattice L is determined by its lattice of subalgebras Sub(L), as well as by its poset of Boolean subalgebras BSub(L). The domain BSub(L) has recently found use in an approach...

This paper is a continuation of the study of the variety generated by the truth value algebra of type-2 fuzzy sets. That variety and some of its reducts were shown to be generated by finite algebras, and in particular to be locally finite. A basic question remaining is whether or not these algebras have finite equational bases, and that is our prin...

The lattice Lu of upper semicontinuous convex normal functions with convolution ordering arises in studies of type-2 fuzzy sets. In 2002, M. Kawaguchi and M. Miyakoshi (5) showed that this lattice is a complete Heyting algebra. Later, J. Harding, C. Walker, and E. Walker (4) gave an improved description of this lattice and showed it was a continuou...

This paper addresses some questions about the variety generated by the algebra of truth values of type-2 fuzzy sets. Its principal result is that this variety is generated by a finite algebra, and in particular is locally finite. This provides an algorithm for determining when an equation holds in this variety. It also sheds light on the question o...

We provide a simple formula to compute the Hausdorff dimension of the attractor of an overlapping iterated function system of contractive similarities satisfying a certain collection of assumptions. This formula is obtained by associating a non-overlapping infinite iterated function system to an iterated function system satisfying our assumptions a...

Let ${\beta(\mathbb{N})}$ denote the Stone–Čech compactification of the set ${\mathbb{N}}$ of natural numbers (with the discrete topology), and let ${\mathbb{N}^\ast}$ denote the remainder ${\beta(\mathbb{N})-\mathbb{N}}$ . We show that, interpreting modal diamond as the closure in a topological space, the modal logic of ${\mathbb{N}^\ast...

Abramsky and Coecke (Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science, pp. 415–425, IEEE Comput. Soc., New York, 2004) have recently introduced an approach to finite dimensional quantum mechanics based on strongly compact closed categories with biproducts. In this note it is shown that the projections of any object A in su...

The algebra of truth values of type-2 fuzzy sets is the set of all functions from the unit interval into itself, with operations defined in terms of certain convolutions of these functions with respect to pointwise max and min. This algebra has been studied rather extensively, both from a theoretical and from a practical point of view. It has a num...

We show that for any infinite cardinal κ, every complete lattice where each element has at most one complement can be regularly embedded into a uniquely complemented κ-complete lattice. This regular embedding preserves all joins and meets, in particular it preserves the bounds of the original lattice. As a corollary, we obtain that every lattice wh...

Ordered algebraic structures are encountered in many areas of mathematics. One frequently wishes to embed a given ordered algebraic structure into a complete ordered algebraic structure in a manner that preserves some aspects of the algebraic and order theoretic properties of the original. It is the purpose here to survey some recent results in thi...

We show that the variety generated by the three-element Heyt-ing algebra admits a meet dense, regular completion even though it is not closed under MacNeille completions.

This fact was rediscovered several times in the 1950's and 1960's (23, 28, 30, 31, 35) and lead to the role of orthomodular lattices (abbreviated: omls) and orthomodular posets (abbreviated: omps) as abstract models for the propositions of a quantum mechanical system. It is instructive to see how the validity of the orthomodular law in C(H) follows...

The algebra of truth values of type-2 fuzzy sets contains isomorphic copies of the algebra of truth values of type-1 fuzzy sets and the algebra of truth values of interval-valued fuzzy sets. The algebra of truth values of type-2 fuzzy sets is not a lattice, but these subalgebras are lattices, and in fact, are complete lattices. There are many other...

For a modal algebra (B,f), there are two natural ways to ex- tend f to an operation on the MacNeille completion of B. The resulting structures are called the lower and upper MacNeille completions of (B,f). In this paper we consider lower and upper MacNeille completions for vari- ous varieties of modal algebras. In particular, we characterize the va...

We show that if a variety V of monotone lattice expansions is finitely generated, then profinite completions agree with canonical extensions on V. The converse holds for varieties of finite type.

We provide a method to construct a type of orthomodular structure known as an orthoalgebra from the direct product decompositions of an object in a category that has finite products and whose ternary product diagrams give rise to certain pushouts. This generalizes a method to construct an orthomodular poset from the direct product decompositions of...

Many algebras arise in the study of fuzzy set theory, including the unit interval with a negation, a t-norm, or both. We investigate equational properties of such algebras.

Let V V be a variety of monotone bounded lattice expansions, that is, bounded lattices endowed with additional operations, each of which is order preserving or reversing in each coordinate. We prove that if V V is closed under MacNeille completions, then it is also closed under canonical extensions. As a corollary we show that in the case of Boolea...

We show there are no non-trivial finite Abelian group-valued measures on the lattice of closed subspaces of an infinite-dimensional Hilbert space, and we use this to establish that the unigroup of the lattice of closed subspaces of an infinite-dimensional Hilbert space is divisible. The main technique is a combinatorial construction of a set of vec...

Let V be a variety of monotone bounded lattice expansions, that is, bounded lattices endowed with additional operations, each of which is order preserving or reversing in each coordinate. We prove that if V is closed under MacNeille completions, then it is also closed under canonical extensions. As a corollary we show that in the case of Boolean al...

An orthomodular lattice (OML) is called concrete if it is isomorphic to a collection of subsets of a set with partial ordering given by set inclusion, orthocomplementation given by set complementation, and finite orthogonal joins given by disjoint unions. Interesting examples of concrete OMLs are obtained by applying Kalmbach''s construction K(L) t...

In this note we provide a topological description of the Mac- Neille completion of a Heyting algebra similar to the description of the Mac- Neille completion of a Boolean algebra in terms of regular open sets of its Stone space. We also show that the only varieties of Heyting algebras that are closed under MacNeille completions are the trivial vari...

We show every at most countable orthomodular lattice is a subalgebra of one generated by three elements. As a corollary we obtain that the free orthomodular lattice on countably many generators is a subalgebra of the free orthomodular lattice on three generators. This answers a question raised by Bruns in 1976 [2] and listed as Problem 15 in Kalmba...

We show every monadic Heyting algebra is isomorphic to a functional monadic Heyting algebra. This solves a 1957 problem of Monteiro and Varsavsky [9].

The notion of a canonical extension of a lattice with additional operations is introduced. Both a concrete description and an abstract characterization of this extension are given. It is shown that this extension is functorial when applied to lattices whose additional operations are either order preserving or reversing, in each coordinate, and vari...

In Harding (Transactions of American Mathematical Society (1996) 348(5), 1839–1862), it was shown that the direct product decompositions of a set X naturally form an orthomodular poset Fact X. Here it is shown that Fact X has a state if and only if X is finite. An example is also given of a finite orthomodular poset that can be embedded into Fact X...

LetP be an orthomodular poset and letB be a Boolean subalgebra ofP. A mappings :P→h 0,1iis said to be a centrally additiveB-state if it is order preserving, satisfies s(a ' ) = 1−s(a), is additive on couples that contain a central element, and restricts to a state onB. It is shown that, for any Boolean subalgebraB ofP,P has an abundance of two-valu...

We prove the result of the title, solving an open problem of Steven Watson (problem 172 in Open Problems in Topology).

We prove a lemma which, under restrictive conditions, shows that epimorphisms inV are surjective if this is true for epimorphisms from irreducible members of V. This lemma is applied to varieties of orthomodular lattices which are generated by orthomodular lattices of bounded height, and to varieties of orthomodular lattices which are generated by...

We prove that, given a nontrivial Boolean algebra B, a compact convex set S and a group G, there is an orthomodular lattice L with the center isomorphic to B, the automorphism group isomorphic to G, and the state space anely homeomorphic to S. Moreover, given an orthomodular lattice J admitting at least one state, L can be chosen such that J is its...

In this survey article we try to give an up-to-date account of certain aspects of the theory of ortholattices (abbreviated OLs), orthomodular lattices (abbreviated OMLs) and modular ortholattices (abbreviated MOLs), not hiding our own research interests. Since most of the questions we deal with have their origin in Universal Algebra, we start with...

Inthispaper a system of axioms ispresented todefine the notion of an experimental system. The primary feature of these axioms is that they are based solely on the mathematical notion of a direct product decomposition of a set. Properties of experimental systems are then developed. This includes defining negation, implication, conjunction, and disju...

In 1996, Harding showed that the binarydecompositions of any algebraic, relational, ortopological structure X form an orthomodular poset FactX. Here, we begin an investigation of the structuralproperties of such orthomodular posets of decompositions.We show that a finite set S of binary decompositions inFact X is compatible if and only if all the b...

Every lattice, and ortholattice, can be represented as the closed elements of some Galois connection on a Boolean algebra. The canonical extension of this Boolean algebra yields a completion of the lattice, or ortholattice. We give a purely order theoretic characterization of this completion, and investigate its properties. While it preserves distr...

We show that the variety of ortholattices has the strong amalgamation property and that the variety of orthomodular lattices has the strong Boolean amalgamation property, i.e. that two orthomodular lattices can be strongly amalgamated over a common Boolean subalgebra. We give examples to show that the variety orthomodular lattices does not have the...

We show that a reducible continuous geometry can be represented as the continuous sections of a bundle of irreducible continuous geometries. We relate this bundle representation to the Pierce sheaf of the continuous geometry and to the subdirect product representation developed by Maeda.

A study is made of Boolean product representations of bounded lattices over the Stone space of their centres. Special emphasis is placed on relating topological properties such as clopen or regular open equalizers to their equivalent lattice theoretic counterparts. Results are also presented connecting various properties of a lattice with propertie...

We present a method of constructing an orthomodular poset from a relation algebra. This technique is used to show that the decompositions of any algebraic, topological, or relational structure naturally form an orthomod-ular poset, thereby explaining the source of orthomodularity in the ortholattice of closed subspaces of a Hilbert space. Several k...