Boolean Group Research Articles

The structure of completely meet irreducible congruences in strongly Fregean algebras

A strongly Fregean algebra is an algebra such that the class of its homomorphic images is Fregean and the variety generated by this algebra is congruence modular. To understand the structure of these algebras we study the prime intervals projectivity relation in the posets of their completely meet irreducible congruences and show that its cosets have the natural structure of a Boolean group. In particular, this approach allows us to represent congruences and elements of such algebras as the subsets of upward closed subsets of these posets with some special properties.

The relevance logic of Boolean groups

Abstract In this article, I consider the positive logic of Boolean groups (i.e. Abelian groups where every non-identity element has order 2), where these are taken as frames for an operational semantics à la Urquhart. I call this logic BG. It is shown that the logic over the smallest nontrivial Boolean group, taken as a frame, is identical to the positive fragment of a quasi-relevance logic that was developed by Robles and Méndez (an extension of this result where negation is included is also discussed). It is proved that BG satisfies the variable sharing property and is Halldén complete, while failing to satisfy the disjunction property. Sound and complete subscripted tableaux are presented for BG. An axiomatization (in the positive language extended with fusion) is presented, which is sound with respect to BG and, with respect to a related ternary relational semantics, complete; the problem of identifying a complete axiomatization for BG itself is left open. Connections between BG and the quasi-relevance logic KR are also discussed.

Centralising Monoids with Low-Arity Witnesses on a Four-Element Set

As part of a project to identify all maximal centralising monoids on a four-element set, we determine all centralising monoids witnessed by unary or by idempotent binary operations on a four-element set. Moreover, we show that every centralising monoid on a set with at least four elements witnessed by the Mal’cev operation of a Boolean group operation is always a maximal centralising monoid, i.e., a co-atom below the full transformation monoid. On the other hand, we also prove that centralising monoids witnessed by certain types of permutations or retractive operations can never be maximal.

$P$-bases and topological groups

A topological space $X$ is defined to have a neighborhood $P$-base at any $x\in X$ from some partially ordered set (poset) $P$ if there exists a neighborhood base $(U_p[x])_{p\in P}$ at $x$ such that $U_p[x]\subseteq U_{p’}[x]$ for all $p\geq p’$ in $P$. We prove that a compact space is countable, hence metrizable, if it has countable scattered height and a $\mathcal {K}(M)$-base for some separable metric space $M$. Banakh [Dissertationes Math. 538 (2019), p. 141] gives a positive answer to Problem 8.6.8. Let $A(X)$ be the free Abelian topological group on $X$. It is shown that if $Y$ is a retract of $X$ such that the free Abelian topological group $A(Y)$ has a $P$-base and $A(X/Y)$ has a $Q$-base, then $A(X)$ has a $P\times Q$-base. Also if $Y$ is a closed subspace of $X$ and $A(X)$ has a $P$-base, then $A(X/Y)$ has a $P$-base. It is shown that any Fréchet-Urysohn topological group with a $\mathcal {K}(M)$-base for some separable metric space $M$ is first-countable, hence metrizable. And if $P$ is a poset with calibre $(\omega _1, \omega )$ and $G$ is a topological group with a $P$-base, then any precompact subset in G is metrizable, hence $G$ is strictly angelic. Applications in function spaces $C_p(X)$ and $C_k(X)$ are discussed. We also give an example of a topological Boolean group of character $\leq \mathfrak {d}$ such that the precompact subsets are metrizable but $G$ doesn’t have an $\omega ^\omega$-base if $\omega _1<\mathfrak {d}$. Gabriyelyan, Kakol, and Liederman [Fund. Math. 229 (2015), pp. 129–158] give a consistent negative answer to Problem 6.5.

Fermionic Topological Order on Generic Triangulations

Consider a finite triangulation of a surface M of genus g and assume that spin-less fermions populate the edges of the triangulation. The quantum dynamics of such particles takes place inside the algebra of canonical anti-commutation relations (CAR). Following Kitaev’s work on toric models, we identify a sub-algebra of CAR generated by elements associated to the triangles and vertices of the triangulation. We show that any Hamiltonian drawn from this sub-algebra displays topological spectral degeneracy. More precisely, if $${{\mathcal {P}}}$$ is any of its spectral projections, the Booleanization of the fundamental group $$\pi _1(M)$$ can be embedded inside the group of invertible elements of the corner algebra $${{\mathcal {P}}}\, \mathrm{CAR} \, {{\mathcal {P}}}$$ . As a consequence, $${{\mathcal {P}}}$$ decomposes in $$4^g$$ lower projections. Furthermore, a projective representation of $${{\mathbb {Z}}}_2^{4g}$$ is also explicitly constructed inside this corner algebra. Key to all these is a presentation of CAR as a crossed product with the Boolean group $$(2^X,\Delta )$$ , where X is the set of fermion sites and $$\Delta $$ is the symmetric difference of its sub-sets.

A Compressed Sensing Approach to Pooled RT-PCR Testing for COVID-19 Detection.

We propose ‘Tapestry’, a single-round pooled testing method with application to COVID-19 testing using quantitative Reverse Transcription Polymerase Chain Reaction (RT-PCR) that can result in shorter testing time and conservation of reagents and testing kits, at clinically acceptable false positive or false negative rates. Tapestry combines ideas from compressed sensing and combinatorial group testing to create a new kind of algorithm that is very effective in deconvoluting pooled tests. Unlike Boolean group testing algorithms, the input is a quantitative readout from each test and the output is a list of viral loads for each sample relative to the pool with the highest viral load. For guaranteed recovery of n}{}k infected samples out of n}{}n gg k being tested, Tapestry needs only n}{}O(k log n) tests with high probability, using random binary pooling matrices. However, we propose deterministic binary pooling matrices based on combinatorial design ideas of Kirkman Triple Systems, which balance between good reconstruction properties and matrix sparsity for ease of pooling while requiring fewer tests in practice. This enables large savings using Tapestry at low prevalence rates while maintaining viability at prevalence rates as high as 9.5%. Empirically we find that single-round Tapestry pooling improves over two-round Dorfman pooling by almost a factor of 2 in the number of tests required. We evaluate Tapestry in simulations with synthetic data obtained using a novel noise model for RT-PCR, and validate it in wet lab experiments with oligomers in quantitative RT-PCR assays. Lastly, we describe use-case scenarios for deployment.

Convergent sequences in topological groups

We survey recent developments concerning the role of convergent sequences in topological groups. We present the Invariant Ideal Axiom and announce its effect on convergence properties in topological groups, in particular, the consistency of the fact that every countable sequential topological group is either metrizable or kω. We also outline a construction of a countably compact topological group without non-trivial convergence sequences using iterated ultrapowers of Bohr topology on a Boolean group from a selective ultrafilter and announce a related ZFC construction of such a group. We recall the main open questions in the area and formulate several new ones.

The impact of the Bohr topology on selective pseudocompactness

Recall that a space X is selectively pseudocompact if for every sequence {Un:n∈N} of non-empty open subsets of X one can choose a point xn∈Un for all n∈N such that the resulting sequence {xn:n∈N} has an accumulation point in X. This notion was introduced under the name strong pseudocompactness by García-Ferreira and Ortiz-Castillo; the present name is due to Dorantes-Aldama and the first listed author. In 2015, García-Ferreira and Tomita constructed a pseudocompact Boolean group that is not selectively pseudocompact. We prove that if the subgroup topology on every countable subgroup H of an infinite Boolean topological group G is finer than its maximal precompact topology (the so-called Bohr topology of H), then G is not selectively pseudocompact, and from this result we deduce that many known examples in the literature of pseudocompact Boolean groups automatically fail to be selectively pseudocompact. We also show that, under the Singular Cardinal Hypothesis, every infinite pseudocompact Boolean group admits a pseudocompact reflexive group topology which is not selectively pseudocompact.

EMBEDDING OF THE FREE ABELIAN TOPOLOGICAL GROUP INTO

We consider the following question: for which metrizable separable spaces does the free abelian topological group isomorphically embed into . While for many natural spaces such an embedding exists, our main result shows that if is a Cook continuum or is a rigid Bernstein set, then does not embed into as a topological subgroup. The analogous statement is true for the free boolean group .

Selectively Pseudocompact Groups without Infinite Separable Pseudocompact Subsets

We give a “naive” (i.e., using no additional set-theoretic assumptions beyond ZFC, the Zermelo-Fraenkel axioms of set theory augmented by the Axiom of Choice) example of a Boolean topological group G without infinite separable pseudocompact subsets having the following “selective” compactness property: For each free ultrafilter p on the set N of natural numbers and every sequence ( U n ) of non-empty open subsets of G, one can choose a point x n ∈ U n for all n ∈ N in such a way that the resulting sequence ( x n ) has a p-limit in G; that is, { n ∈ N : x n ∈ V } ∈ p for every neighbourhood V of x in G. In particular, G is selectively pseudocompact (strongly pseudocompact) but not selectively sequentially pseudocompact. This answers a question of Dorantes-Aldama and the first listed author. The group G above is not pseudo- ω -bounded either. Furthermore, we show that the free precompact Boolean group of a topological sum ⨁ i ∈ I X i , where each space X i is either maximal or discrete, contains no infinite separable pseudocompact subsets.

J-Boolean group rings and skew group rings

A ring [Formula: see text] is called semiboolean if [Formula: see text] is boolean and idempotents lift modulo [Formula: see text], where [Formula: see text] denotes the Jacobson radical of [Formula: see text]. In this paper, we define [Formula: see text]-boolean rings as a generalization of semiboolean rings. A ring [Formula: see text] is said to be J-boolean if [Formula: see text] is boolean. Various basic properties of these rings are obtained. The [Formula: see text]-boolean group rings and skew group rings have been studied. It is investigated whether the results obtained for [Formula: see text]-boolean group rings also hold for the skew group rings.

Duality properties of bounded torsion topological abelian groups

Let G be a precompact, bounded torsion abelian group and Gp∧ its dual group endowed with the topology of pointwise convergence. We prove that if G is Baire (resp., pseudocompact), then all compact (resp., countably compact) subsets of Gp∧ are finite. We also prove that G is pseudocompact if and only if all countable subgroups of Gp∧ are closed. We present other characterizations of pseudocompactness and the Baire property of Gp∧ in terms of properties that express in different ways the abundance of continuous characters of G. Besides, we give an example of a precompact boolean group G with the Baire property such that the dual group Gp∧ contains an infinite countably compact subspace without isolated points.

Strongly Summable Ultrafilters, Union Ultrafilters, and the Trivial Sums Property

AbstractWe answer two questions of Hindman, Steprāns, and Strauss; namely, we prove that every strongly summable ultrafilter on an abelian group is sparse and has the trivial sums property. Moreover, we show that in most cases the sparseness of the given ultrafilter is a consequence of its being isomorphic to a union ultrafilter. However, this does not happen in all cases; we also construct (assuming Martin's Axiom for countable partial orders, i.e., , a strongly summable ultrafilter on the Boolean group that is not additively isomorphic to any union ultrafilter.

Ultrasonography predicts achievement of Boolean remission after DAS28-based clinical remission of rheumatoid arthritis

Objectives. To determine whether ultrasonography (US) predicts Boolean remission in rheumatoid arthritis (RA) patients who had achieved disease activity score in 28 joints (DAS28)-based remission criteria.Methods. Thirty-one RA patients in DAS28-based clinical remission were recruited. US semiquantitatively determined Gray scale (GS) and power Doppler (PD) signal scores in the bilateral wrists and all metacarpophalangeals and proximal interphalangeals. Total GS score and total PD score were calculated as the sum of individual scores for each joint.Results. Among 22 RA patients, who maintained DAS28 remission for 2 years, 16 met Boolean remission criteria at the end of study. Both total GS and total PD scores at baseline were significantly lower in Boolean remission group than non-remission group. There was no significant difference in other baseline parameters, including duration of disease, duration of remission, mTSS, and disease activity composite parameters between the two groups. Among the factors for Boolean remission criteria at 2 years, patient global assessment score was associated with total GS score at the entry, while swollen joint count was related to total PD score.Conclusions. Null or low grade of GS and PD findings in US are associated with achieving Boolean remission. Thus, US is essential for assessment and prediction of “deeper remission” of RA.

Notes on non-archimedean topological groups

We show that the Heisenberg type group HX=(Z2⊕V)⋋V⁎, with the discrete Boolean group V:=C(X,Z2), canonically defined by any Stone space X, is always minimal. That is, HX does not admit any strictly coarser Hausdorff group topology. This leads us to the following result: for every (locally compact) non-archimedean G there exists a (resp., locally compact) non-archimedean minimal group M such that G is a group retract of M. For discrete groups G the latter was proved by S. Dierolf and U. Schwanengel (1979) [6]. We unify some old and new characterization results for non-archimedean groups.

Self-duality in the class of precompact groups

A topological Abelian group G is called ( strongly) self-dual if there exists a topological isomorphism Φ : G → G ∧ of G onto the dual group G ∧ (such that Φ ( x ) ( y ) = Φ ( y ) ( x ) for all x , y ∈ G ). We prove that every countably compact self-dual Abelian group is finite. It turns out, however, that for every infinite cardinal κ with κ ω = κ , there exists a pseudocompact, non-compact, strongly self-dual Boolean group of cardinality κ.

On Closed Ideals of βG

For every discrete group G, the Stone-Cech compactification βG has a natural structure of a right topological semigroup. Using the concept of Boolean group ideal on G, we propose a way of constructing closed ideals of the semigroup βG. In particular, we show that, for every infinite Abelian group G of cardinality κ, there are \(2^{2^{\kappa}}\) distinct closed ideals in βG.

A systematic data integration technique for mold-surface polishing processes planning

This paper presents a new MSDI (mold surface data integration) method for presenting ruled mold-surface data for CAPP applications. The method makes use of adjacency and feature-data matrices to establish a framework of feature-based data. The MSDI method enables the storage, retrieval and operations of geometries, topologies, and machining information of a given ruled mold surface. The matrices of adjacency and feature-data effectively represent the relations. Boolean algebra and group technology (GT) methods are introduced to generate the desired process plans. A sample mold made of ruled surfaces is made and used to show the effectiveness of the proposed data representation and the integrated mold surface processing method. A compact computer program is implemented to enhance the speed and accuracy of the MSDI method. The numerical results indicate that the mold surface decomposition and integration procedure presented in this paper are systematic and effective. The paper provides a novel modular mold surface modeling tool and procedure that can be further extended to accommodate more features appearing in complicated ruled mold surfaces.

Square of countably compact groups without non-trivial convergent sequences

We show in ZFC that the existence of a countably compact Abelian group without non-trivial convergent sequences implies the existence of a countably compact group whose square is not countably compact. This improves a result obtained by van Douwen in 1980: the existence of a countably compact Boolean group without non-trivial convergent sequences implies the existence of two countably compact groups whose product is not countably compact in ZFC. Hart and van Mill showed in 1991 the existence of a countably compact group whose square is not countably compact under Martin's Axiom for countable posets. We show that the existence of such an example does not depend on some form of Martin's Axiom.

Finitely generated varieties of distributive double p-algebras universal modulo a group

For any monoid M, any universal variety contains arbitrarily large algebras whose endomorphism monoid is isomorphic to M. A variety universal modulo a group G contains arbitrarily large algebras whose endomorphism monoid is isomorphic to the direct product M x G. One of the results of this paper structurally characterizes all finitely generated varieties of distributive double p-algebras universal modulo a group, and shows that any unavoidable direct factor G is a Boolean group with at most eight elements.