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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., Comment: 18 pages, 4 figures

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., Comment: 6 pages, 5 figures

In 1973, Katri\v{n}\'{a}k 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 Katri\v{n}\'{a}k's theorem, in the sense that in the variety RDPCH of regular dually pseudocomplemented Heyting algebras, the implication operation $\to$ satisfies the Katri\v{n}\'{a}k's formula. As applications of this result together with the above-mentioned Katri\v{n}\'{a}k'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 $2^{\aleph_0}$., Comment: 16 pages

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., Comment: 7 figures, accepted for publication in J. Phys. B

Nelson algebras are defined in [Diana Brignole and Ant\'onio Monteiro. Caract\'erisation des alg\`ebres de Nelson par des \'egalit\'es. {I}, {II}. Proc. Japan Acad., 43:279--283; 284--285, 1967.] in terms of the language $\langle \land,\lor,\to,\sim,1\rangle$. In 1962, Diana Brignole [Diana Brignole. Axiomatizaci\'on de un $N$-reticulado. Revista de la Uni\'on Matem\'atica Argentina, XXI(3):147--148, 1965.], solved the problem proposed by Antonio Monteiro, of giving an axiomatization of Nelson algebras in terms of the connectives $\rightarrowtail$, $\land$ and the constant $0=\sim 1$. In this work, we present for the first time a complete proof of this fact, and also show the dependence and independence of some of the axioms proposed by Brignole., Comment: 28 pages

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., Comment: Accepted for publication in New Journal of Physics

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., Comment: 11 pages

We present a category equivalent to that of semi-Nelson algebras. The objects in this category are pairs consisting of a semi-Heyting algebra and one of its filters. The filters must contain all the dense elements of the semi-Heyting algebra and satisfy an additional technical condition. We also show that in the case of dually hemimorphic semi-Nelson algebras, the filters are not necessary and the category is equivalent to that of dually hemimorphic semi-Heyting algebras.

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., Comment: 12 pages

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., Comment: 10 pages, 4 figures, 1 table; published version

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., Comment: 53 pages, 5 figures

Cooling and detection schemes using laser cooling and methods of quantum logic can contribute to high precision CPT symmetry tests in the baryonic sector. This work introduces an experiment to sympathetically cool protons and antiprotons using the Coulomb interaction with a $^9$Be$^+$ ion trapped in a nearby but separate potential well. We have designed and set up an apparatus to show such coupling between two identical ions for the first time in a Penning trap. In this paper, we present evidence for successful loading and Doppler cooling of clouds and single ions. Our coupling scheme has applications in a range of high-precision measurements in Penning traps and has the potential to substantially improve motional control in these experiments., Comment: Corrected typos

The Penning-trap electronic-detection technique that offers the precision and sensitivity requested in mass spectrometry for fundamental studies in nuclear and particle physics has not been proven yet to be universal. This has motivated the construction of a Penning-trap facility aiming at the implementation of a novel detection method, consisting in measuring motional frequencies of singly-charged trapped ions in strong magnetic fields, through the fluorescence photons from the 4s$^2$S$_{1/2}\rightarrow $4p$^2$P$_{1/2}$ atomic transition in $^{40}$Ca$^+$. The key element of this facility is an open-ring Penning trap, built and fully characterized, which is coupled upstream to a preparation Penning trap similar to those built at Radioactive Ion Beam facilities. Motional frequency measurements of trapped ions stored in the open-ring trap have been carried out by applying external dipolar and quadrupolar fields in resonance with the ions' eigenmotions, in combination with time-of-flight identification. The infrastructure to observe the fluorescence photons from $^{40}$Ca$^+$, comprising the twelve laser beams needed in 7~Tesla, and a two-meters long system to register the image in a high-sensitive CCD sensor, has also been successfully tested by observing optically the trapped $^{40}$Ca$^+$ ions. This demonstrates the capabilities of this facility for the proposed laser-based mass-spectrometry technique, and introduces it as a unique platform to perform laser-spectroscopy experiments with implications in different fields of physics.

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., Comment: 33 pages. arXiv admin note: text overlap with arXiv:1710.10559, arXiv:1710.10408

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 $3$., Comment: 29 pages. arXiv admin note: text overlap with arXiv:1710.10408

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 $x' : = x \to 0$. An implication 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 $\mathcal S$. We began a systematic analysis of weak associative laws of length $\leq 4$ in [CS16e], by examining the identities of Bol-Moufang type in the context of the variety $\mathcal S$. In this paper we complete the analysis by investigating the rest of the weak associative laws of length $\leq 4$ relative to $\mathcal S$. We show that, of the 155 subvarieties of $\mathcal S$ defined by the weak associative laws of size $\leq 4$, there are exactly $6$ distinct ones. We also give an explicit description of the poset of the (distinct) subvarieties of $\mathcal S$ defined by weak associative laws of length $\leq 4$., Comment: 36 pages

We provide a recursive construction of all the semi-Heyting algebras that can be defined on a chain with $n$ elements. This construction allows us to count them easily. We also compare the formula for the number of semi-Heyting chains thus obtained to the one previously known.

We offer the proofs that complete our article introducing the propositional calculus called semi-intuitionistic logic with strong negation., Comment: Contains proofs omitted, because of their extention, from an article published in Studia Logica

We have studied the dynamics of one and two laser-cooled trapped $^{40}$Ca$^+$ ions by applying electric fields of different nature along the axial direction of the trap, namely, driving the motion with a harmonic dipolar field, or with white noise. These two types of driving induce distinct motional states of the axial modes; a coherent oscillation with the dipolar field, or an enhanced Brownian motion due to an additional contribution to the heating rate from the electric noise. In both scenarios, the sensitivity of an isolated ion and a laser-cooled two-ion crystal has been evaluated and compared. The analysis and understanding of this dynamics is important towards the implementation of a novel Penning-trap mass-spectroscopy technique based on optical detection, aiming at improving precision and sensitivity., Comment: 13 pages, 9 figures, Submitted on the 30th of June to the Special Issue: Quantum optics, cooling and collisions of ions and atoms (references have been updated from the submitted version)

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 $n$., Comment: 35 pages

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., Comment: 21 pages

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., Comment: This paper, except the appendix, will appear in Algebra Universalis. 25 pages, 4 figures, a revised version with a new title