arXiv subjects

Juan M. Cornejo

Publications and source records attributed to Juan M. Cornejo.

17 recordsLinked to original sources

arXiv2026

On the potential for high-accuracy spectroscopy of $\mathrm{H}_2^+$ and $\overline{\mathrm{H}}_2^-$ in Penning traps for a test of CPT invariance

The comparison of vibrational transition frequencies of $\mathrm{H}_2^+$ and $\overline{\mathrm{H}}_2^-$ offers a new opportunity to test CPT invariance. Myers [Phys. Rev. A 98, 010101(R) (2018)] proposed performing laser spectroscopy in a Penning trap (PT) with non-destructive read-out. Here, we provide an extensive analysis of this proposal, introduce novel aspects, and discuss its implementation in PTs that incorporate either the continuous Stern-Gerlach effect or quantum-logic spectroscopy. We derive estimates for the achievable accuracy of the test. We find that a comparison of the vibrational frequencies at a fractional level of $1\times10^{-17}$ is a realistic prospect, using technology that is mostly already available. We also analyze complementary CPT invariance tests, namely those of the g-factor of the bound electron/positron via electron-spin-resonance spectroscopy and of the magnetic moment of the proton/antiproton via radiofrequency spectroscopy.

quant-ph↗

arXiv2025

Connexive logics and connexive semi-Heyting algebras

In this paper, we define and investigate a connexive logic, called 'Connexive semi-Heyting logic' (\mathcal{CSH} for short) and a new subvariety CSH of the variety SH of semi-Heyting algebras. It is shown that the logic \mathcal{CSH} is implicative in the sense of Rasiowa, and is algebraizable with CSH as an equivalent algebraic semantics (in the sense of Blok and Pigozzi). We also introduce the logics \mathcal{AT}i and \mathcal{BT}i, i = 1, 2, along with the subvarieties ATi and BTi, i = 1, 2, of SH. It is then shown that AT1 = AT2 and CSH = BT1 \subset BT2 \subset AT1. A 3-valued connexive semi-Heyting logic \mathcal{CSH}3 and its equivalent algebraic semantics CSH3 are introduced and axiomatized; and it is then shown that CSH3 is deductively equivalent to the 3-valued intuitionistic logic. New characterizations of anti-Boolean semi-Heyting algebras are given. We show that BT2 \cap SHc = V(2), and SHc \subset AT1, where SHc is defined by x \to y = y \to x. It is proved that the identity (AT1) is equivalent to the identity x* \to y* = y* \to x* (* being the pseudocomplement) in StSH and also is equivalent to 0 \to 1 = 0 in SH. We show that AT1 \cap EX \subset BT1, where EX is defined by x \to (y \to z) = y \to (x \to z). The paper concludes with some further remarks, mentions some open problems for future research and proposes two new principles to be considered as Connexive Theses.

math.LO↗

arXiv2024

Amalgamation Property in the subvarieties of Gautama and Almost Gautama algebras

Gautama algebras were introduced recently, as a common generalization of regular double Stone algebras and regular Kleene Stone algebras. Even more recently, Gautama algebras were further generalized to Almost Gautama algebras (AG for short). The main purpose of this paper is to investigate the Amalgamation Property (AP, for short) in the subvarieties of the variety AG. In fact, we show that, of the eight nontrivial subvarieties of AG, only four varieties, namely those of Boolean algebras, of regular double Stone algebras, of regular Kleene Stone algebras and of De Morgan Boolean algebras have the AP and the remaining four do not have the AP. We give several applications of this result; in particular, we examine the following properties for the subvarieties of AG: transferability property (TP), having enough injectives (EI), Embedding Property, Bounded Obstruction Property and having a model companion.

math.LO↗

arXiv2023

Resolved-sideband cooling of a single $^9$Be$^+$ ion in a Penning trap

Manipulating individual trapped ions at the single quantum level has become standard practice in radio-frequency ion traps, enabling applications from quantum information processing to precision metrology. The key ingredient is ground-state cooling of the particle's motion through resolved-sideband laser cooling. Ultra-high-presicion experiments using Penning ion traps will greatly benefit from the reduction of systematic errors offered by full motional control, with applications to atomic masses and $g$-factor measurements, determinations of fundamental constants or related tests of fundamental physics. In addition, it will allow to implement quantum logic spectroscopy, a technique that has enabled a new class of precision measurements in radio-frequency ion traps. Here we demonstrate resolved-sideband laser cooling of the axial motion of a single $^9$Be$^+$ ion in a cryogenic 5 Tesla Penning trap system using a two-photon stimulated-Raman process, reaching a mean phonon number of $\bar{n}_z = 0.10(4)$. This is a fundamental step in the implementation of quantum logic spectroscopy for matter-antimatter comparison tests in the baryonic sector of the Standard Model and a key step towards improved precision experiments in Penning traps operating at the quantum limit.

physics.atom-ph↗

arXiv2022

Regular Double $p$-Algebras: A converse to a Katriňák's Theorem, and Applications

In 1973, Katriňák proved that regular double $p$-algebras can be regarded as (regular) double Heyting algebras by ingeniously constructing binary terms for the Heying implication and its dual in terms of pseudocomplement and its dual. In this paper we prove a converse to the Katriňák's theorem, in the sense that in the variety RDPCH of regular dually pseudocomplemented Heyting algebras, the implication operation $\to$ satisfies the Katriňák's formula. As applications of this result together with the above-mentioned Katriňák's theorem, we show that the varieties RDBLP, RDPCH, RPCH$^d$ and RDBLH of regular double $p$-algebras, regular dually pseudocomplemented Heyting algebras, regular pseudocomplemented dual Heyting algebras, and regular double Heyting algebras, respectively, are term-equivalent to each other and also that the varieties RDMP, RDMH, RDMDBLH, RDMDBLP of regular De Morgan $p$-algebras, regular De Morgan Heyting algebras, regular De Morgan double Heyting algebras, and regular De Morgan double $p$-algebras, respectively, are also term equivalent to each other. From these results and recent results of Adams, Sankappanavar and vaz de Carvalho, we deduce that the lattices of subvarieties of all these varieties have cardinality $2^{\aleph_0}$. We then define new logics, RDPCH, RPCHd, and RDMH, and show that they are algebraizable with RDPCH, RPCH$^d$ and RDMH, respectively as their equivalent algebraic semantics. It is also deduced that the lattices of extensions of all of the above mentioned logics have cardinality $2^{\aleph_0}$.

math.LO↗

arXiv2022

A Logic for Dually Hemimorphic Semi-Heyting Algebras and its Axiomatic Extensions

In this paper, we focus on the variety DHMSH of dually hemimorphic semi-Heyting algebras from a logical point of view. Firstly, we present a Hilbert-style axiomatization of a new logic called Dually hemimorphic semi-Heyting logic (DHMSH, for short), as an expansion of semi-intuitionistic logic SI (also called SH) introduced by the first author by adding a weak negation (to be interpreted as a dual hemimorphism). We then prove that it is implicative in the sense of Rasiowa and that it is complete with respect to the variety DHMSH. It is deduced that the logic DHMSH is algebraizable in the sense of Blok and Pigozzi, with the variety DHMSH as its equivalent algebraic semantics and that the lattice of axiomatic extensions of DHMSH is dually isomorphic to the lattice of subvarieties of DHMSH. A new axiomatization for Moisil's logic is also obtained. Secondly, we characterize the axiomatic extensions of DHMSH in which the Deduction Theorem holds. Thirdly, we present several new logics, extending the logic DHMSH, corresponding to several important subvarieties of the variety DHMSH. These include logics corresponding to the varieties generated by two-element, three-element and some four-element dually quasi-De Morgan semi-Heyting algebras, as well as a new axiomatization for the 3-valued Lukasiewiczlogic. Surprisingly, many of these logics turn out to be connexive logics, a few of which are presented in this paper. Fourthly, we present axiomatizations for two infinite sequences of logics namely, De Morgan-Goedel logics and dually pseudocomplemented Goedel logics, Fifthly, axiomatizations are also provided for logics corresponding to many subvarieties of regular dually quasi-De Morgan Stone semi-Heyting algebras, of regular De Morgan semi-Heyting algebras of level 1, and of JI-distributive semi-Heyting algebras of level 1. We conclude the paper with some open problems.

math.LO↗

arXiv2021

139 GHz UV phase-locked Raman laser system for thermometry and sideband cooling of $^9$Be$^+$ ions in a Penning trap

We demonstrate phase locking of two ultraviolet laser sources by modulating a fundamental infrared laser with 4th-order sidebands using an electro-optic modulator and phase locking of one sideband to a second fundamental infrared laser. Subsequent sum frequency generation and second harmonic generation successfully translates the frequency offset to the ultraviolet domain. The phase lock at 139 GHz is confirmed through stimulated Raman transitions for thermometry of $^9$Be$^+$ ions confined in a cryogenic Penning trap. This technique might be used for sideband cooling of single $^9$Be$^+$ ions as well as sympathetic cooling schemes and quantum logic based measurements in Penning traps in the future.

physics.atom-ph↗

arXiv2021

Quantum logic inspired techniques for spacetime-symmetry tests with (anti-)protons

Cosmological observations as well as theoretical approaches to physics beyond the Standard Model provide strong motivations for experimental tests of fundamental symmetries, such as CPT invariance. In this context, the availability of cold baryonic antimatter at CERN has opened an avenue for ultrahigh-precision comparisons of protons and antiprotons in Penning traps. This work discusses an experimental method inspired by quantum logic techniques that will improve particle localization and readout speed in such experiments. The method allows for sympathetic cooling of the (anti-)proton to its quantum-mechanical ground state as well as the readout of its spin alignment, replacing the commonly used continuous Stern-Gerlach effect. Both of these features are achieved through coupling to a laser-cooled `logic' ion co-trapped in a double-well potential. This technique will boost the measurement sampling rate and will thus provide results with lower statistical uncertainty, contributing to stringent searches for time dependent variations in the data. Such measurements ultimately yield extremely high sensitivities to CPT violating coefficients acting on baryons in the Standard-Model Extension, will allow the exploration of previously unmeasured types of symmetry violations, and will enable antimatter-based axion-like dark matter searches with improved mass resolution.

hep-ph↗

arXiv2020

Semidistributivity and Whitman Property in Implication Zroupoids

In 2012, the second author introduced and studied the variety $\mathcal{I}$ of implication zroupoids that generalize De Morgan algebras and $\lor$-semilattices with $0$. An algebra $\mathbf A = \langle A, \to, 0 \rangle$, where $\to$ is binary and $0$ is a constant, is called an \emph{implication zroupoid} ($\mathcal{I}$-zroupoid, for short) if $\mathbf A$ satisfies: $(x \to y) \to z \approx [(z' \to x) \to (y \to z)']'$, where $x' : = x \to 0$, and $ 0'' \approx 0$. Let $\mathcal{I}$ denote the variety of implication zroupoids and $\mathbf A \in \mathcal{I}$. For $x,y \in \mathbf A$, let $x \land y := (x \to y')'$ and $x \lor y := (x' \land y')'$. In an earlier paper we had proved that if $\mathbf A \in \mathcal{I}$, then the algebra $\mathbf A_{mj} = \langle A, \lor, \land \rangle$ is a bisemigroup. In this paper we generalize the notion of semi-distributivity from lattices to bisemigroups and prove that, for every $\mathbf A \in \mathcal{I}$, the bisemigroup $\mathbf A_{mj}$ is semidistributive. Secondly, we generalize the Whitman Property from lattices to bisemigroups and prove that the subvariety $\mathcal{MEJ}$ of $\mathcal I$, defined by the identity: $x \land y \approx x \lor y$, satisfies the Whitman Property.

math.LO↗

arXiv2020

Elementary laser-less quantum logic operations with (anti-)protons in Penning traps

Static magnetic field gradients superimposed on the electromagnetic trapping potential of a Penning trap can be used to implement laser-less spin-motion couplings that allow the realization of elementary quantum logic operations in the radio-frequency regime. An important scenario of practical interest is the application to $g$-factor measurements with single (anti-)protons to test the fundamental charge, parity, time reversal (CPT) invariance as pursued in the BASE collaboration [Smorra et al., Eur. Phys. J. Spec. Top. 224, 3055-3108 (2015), Smorra et al., Nature 550, 371-374 (2017), Schneider et al., Science 358, 1081-1084 (2017)]. We discuss the classical and quantum behavior of a charged particle in a Penning trap with a superimposed magnetic field gradient. Using analytic and numerical calculations, we find that it is possible to carry out a SWAP gate between the spin and the motional qubit of a single (anti-)proton with high fidelity, provided the particle has been initialized in the motional ground state. We discuss the implications of our findings for the realization of quantum logic spectroscopy in this system.

physics.atom-ph↗

arXiv2020

Implication Zroupoids and Birkhoff Systems

An algebra $A = \langle A, \to, 0 \rangle$, where $\to$ is binary and $0$ is a constant, is called an implication zroupoid (I-zroupoid, for short) if A satisfies the identities: $(x \to y) \to z \approx ((z' \to x) \to (y \to z)')'$, where $x' := x \to 0$, and $0'' \approx 0$. These algebras generalize De Morgan algebras and $\lor$-semilattices with zero. Let I denote the variety of implication zroupoids. For details on the motivation leading to these algebras, we refer the reader to [San12] (or the relevant papers mentioned at the end of this paper). The investigations into the structure of the lattice of subvarieties of I, begun in [San12], have continued in [CS16a, CS16b, CS17a, CS17b, CS18a, CS18b, CS19] and [GSV19]. The present paper is a sequel to this series of papers and is devoted to making further contributions to the theory of implication zroupoids. The identity (BR): $x \land (x \lor y) \approx x \lor (x \land y)$ is called the Birkhoff's identity. The main purpose of this paper is to prove that if A is an algebra in the variety I, then the derived algebra $A_{mj} := \langle A; \land, \lor \rangle$, where $a \land b := (a \to b')'$ and $a \lor b := (a' \land b')'$, satisfies the Birkhoff's identity. As a consequence, we characterize the implication zroupoids A whose derived algebras $A_{mj}$ are Birkhoff systems. It also follows from the main result that there are bisemigroups that are not bisemilattices but satisfy the Birkhoff's identity, which suggests a more general notion, than Birkhoff systems, of "Birkhoff bisemigroups" as bisemigroups satisfying the Birkhoff's identity. The paper concludes with an open problem on Birkhoff bisemigroups.

math.LO↗

arXiv2017

Implication Zroupoids and Identities of Associative Type

An algebra $\mathbf A = \langle A, \to, 0 \rangle$, where $\to$ is binary and $0$ is a constant, is called an implication zroupoid ($\mathcal I$-zroupoid, for short) if $\mathbf A$ satisfies the identities: $(x \to y) \to z \approx [(z' \to x) \to (y \to z)']'$ and $ 0'' \approx 0$, where $x' : = x \to 0$, and $\mathcal I$ denotes the variety of all $\mathcal I$-zroupoids. An $\mathcal I$-zroupoid is symmetric if it satisfies $x'' \approx x$ and $(x \to y')' \approx (y \to x')'$. The variety of symmetric $\mathcal I$-zroupoids is denoted by $\mathcal S$. An identity $p \approx q$, in the groupoid language $\langle \to \rangle$, is called an identity of associative type of length $3$ if $p$ and $q$ have exactly 3 (distinct) variables, say x,y,z, and are grouped according to one of the two ways of grouping: (1) $\star \to (\star \to \star)$ and (2) $(\star \to \star) \to \star$, where $\star$ is a place holder for a variable. A subvariety of $\mathcal I$ is said to be of associative type of length $3$, if it is defined, relative to $\mathcal I$, by a single identity of associative type of length $3$. In this paper we give a complete analysis of the mutual relationships of all subvarieties of $\mathcal I$ of associative type of length $3$. We prove, in our main theorem, that there are exactly 8 such subvarieties of $\mathcal I$ that are distinct from each other and describe explicitly the poset formed by them under inclusion. As an application of the main theorem, we derive that there are three distinct subvarieties of the variety $\mathcal S$, each defined, relative to $\mathcal S$, by a single identity of associative type of length $3$.

math.LO↗

arXiv2017

Symmetric implication zroupoids and identities of Bol-Moufang type

An algebra $\mathbf A = \langle A, \to, 0 \rangle$, where $\to$ is binary and $0$ is a constant, is called an implication zroupoid ($\mathcal I$-zroupoid, for short) if $\mathbf A$ satisfies the identities: (I): $(x \to y) \to z \approx ((z' \to x) \to (y \to z)')'$, and (I$_{0}$): $ 0'' \approx 0$, where $x' : = x \to 0$. An implication zroupoid is symmetric if it satisfies the identities: $x'' \approx x$ and $(x \to y')' \approx (y \to x')'$. An identity is of Bol-Moufang type if it contains only one binary operation symbol, one of its three variables occurs twice on each side, each of the other two variables occurs once on each side, and the variables occur in the same (alphabetical) order on both sides of the identity. In this paper we make a systematic analysis of all $ 60$ identities of Bol-Moufang type in the variety $\mathcal S$ of symmetric $\mathcal I$-zroupoids. We show that $47$ of the subvarieties of $\mathcal S$, defined by the identities of Bol-Moufang type are equal to the variety $\mathcal{SL}$ of $\lor$-semilattices with the least element $0$ and, one of the others is equal to $\mathcal S$. Of the remaining 12, there are only $3$ distinct ones. We also give an explicit description of the poset of the (distinct) subvarieties of $\mathcal S$ of Bol-Moufang type.

math.LO↗

arXiv2017

Symmetric Implication Zroupoids and Weak Associative Laws

math.LO↗

arXiv2016

On Implicator Groupoids

In a paper published in 2012, the second author extended the well-known fact that Boolean algebras can be defined using only implication and a constant, to De Morgan algebras-this result led him to introduce, and investigate (in the same paper), the variety I of algebras, there called implication zroupoids (I-zroupoids) and here called implicator gruopids (I- groupoids), that generalize De Morgan algebras. The present paper is a continuation of the paper mentioned above and is devoted to investigating the structure of the lattice of subvarieties of I, and also to making further contributions to the theory of implicator groupoids. Several new subvarieties of I are introduced and their relationship with each other, and with the subvarieties of I which were already investigated in the paper mentioned above, are explored.

math.LO↗

arXiv2015

Order in Implication Zroupoids

The variety $\mathbf{I}$ of implication zroupoids was defined and investigated by Sankappanavar ([7]) as a generalization of De Morgan algebras. Also, in [7], several new subvarieties of $\mathbf{I}$ were introduced, including the subvariety $\mathbf{I_{2,0}}$, defined by the identity: $x" \approx x$, which plays a crucial role in this paper. Several more new subvarieties of $\mathbf{I}$, including the subvariety $\mathbf{SL}$ of semilattices with a least element $0$, are studied in [3], and an explicit description of semisimple subvarieties of $\mathbf{I}$ is given in [5]. It is well known that the operation $\land$ induces a partial order ($\sqsubseteq$) in the variety $\mathbf{SL}$ and also in the variety $\mathbf{DM}$ of De Morgan algebras. As both $\mathbf{SL}$ and $\mathbf{DM}$ are subvarieties of $\mathbf{I}$ and the definition of partial order can be expressed in terms of the implication and the constant, it is but natural to ask whether the relation $\sqsubseteq$ (now defined) on $\mathbf{I}$ is actually a partial order in some (larger) subvariety of $\mathbf{I}$ that includes $\mathbf{SL}$ and $\mathbf{DM}$. The purpose of the present paper is two-fold: Firstly, a complete answer is given to the above mentioned problem. Indeed, our first main theorem shows that the variety $\mathbf{I_{2,0}}$ is a maximal subvariety of $\mathbf{I}$ with respect to the property that the relation $\sqsubseteq$ is a partial order on its members. In view of this result, one is then naturally led to consider the problem of determining the number of non-isomorphic algebras in $\mathbf{I_{2,0}}$ that can be defined on an $n$-element chain (herein called $\mathbf{I_{2,0}}$-chains), $n$ being a natural number. Secondly, we answer this problem in our second main theorem, which says that, for each $n \in \mathbb{N}$, there are exactly $n$ nonisomorphic $\mathbf{I_{2,0}}$-chains of size $n$.

math.LO↗

arXiv2015

Semisimple Varieties of Implication Zroupoids

It is a well known fact that Boolean algebras can be defined using only implication and a constant. In 2012, this result was extended to De Morgan algebras in [8] which led Sankappanavar to introduce, and investigate, the variety I of implication zroupoids generalizing De Morgan algebras. His investigations were continued in [3] and [4] in which several new subvarieties of I were introduced and their relationships with each other and with the varieties of [8] were explored. The present paper is a continuation of [8] and [3]. The main purpose of this paper is to determine the simple algebras in I. It is shown that there are exactly five simple algebras in I. From this description we deduce that the semisimple subvarieties of I are precisely the subvarieties of the variety generated by these 5 simple I-zroupoids and are locally finite. It also follows that the lattice of semisimple subvarieties of I is isomorphic to the direct product of a 4-element Boolean lattice and a 4-element chain.

math.LO↗