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.Comment: 21 page

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 $n$.Comment: 35 page

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

Compton Scattering Polarimetry for The Determination of the Proton’S Weak Charge Through Measurements of the Parity-Violating Asymmetry of 1H(E,e\u27)P

The Standard Model has been a theory with the greatest success in describing the fundamental interactions of particles. as of the writing of this dissertation, the Standard Model has not been shown to make a false prediction. However, the limitations of the Standard Model have long been suspected by its lack of a description of gravity, nor dark matter. its largest challenge to date, has been the observation of neutrino oscillations, and the implication that they may not be massless, as required by the Standard Model. The growing consensus is that the Standard Model is simply a lower energy effective field theory, and that new physics lies at much higher energies. The Qweak Experiment is testing the Electroweak theory of the Standard Model by making a precise determination of the weak charge of the proton (Qpw). Any signs of “new physics” will appear as a deviation to the Standard Model prediction. The weak charge is determined via a precise measurement of the parity-violating asymmetry of the electron-proton interaction via elastic scattering of a longitudinally polarized electron beam of an un-polarized proton target. The experiment required that the electron beam polarization be measured to an absolute uncertainty of 1 %. at this level the electron beam polarization was projected to contribute the single largest experimental uncertainty to the parity-violating asymmetry measurement. This dissertation will detail the use of Compton scattering to determine the electron beam polarization via the detection of the scattered photon. I will conclude the remainder of the dissertation with an independent analysis of the blinded Qweak

A Quantum Sensor for Neutrino Mass Measurements

There are few experiments aiming at determining directly the mass of the electron antineutrino with a sensitivity of 0.2 eV by analyzing the end of the -decay spectrum of specific nuclei. This sensitivity can be only reached if the uncertainties arising from systematic effects are very small and very well determined. The same holds for experiments aiming at improving the sensitivity in the determination of the mass of the electron neutrino using electron-capture ()-decaying nuclei. One important input in these cases is an accurate Q-value of the decay which can be unambiguously determined from the difference of the mass of the mother and the daughter nuclei by means of Penning traps. In order to reach the required sensitivity, a novel device called Quantum Sensor is under construction at the University of Granada (Spain). The device will allow measuring atomic masses, and therefore Q-values from decays with unprecedented accuracy and sensitivity, using fluorescence photons from a laser-cooled ion instead of electronic detection. This paper will give an overview on Q-value measurements performed with Penning traps, relevant for neutrino mass spectrometry, describing the Quantum Sensor and the facility under construction. It will end by presenting the status of the project.The construction of the device described in this paper has been recently started and it is funded by the European Research Council within the ERC-2011-StG call (contract no. 268648-TRAPSENSOR). Besides the applications for neutrino mass spectrometry, the device has been also conceived for applications in the field of nuclear physics. During the conception of the project, D. Rodríguez acknowledges funding from the Spanish Ministry of Science and Innovation (now integrated in the Ministry for Economy and Competitiveness) through the projects FPA2009-14091-C02-02 and FPA2010-14803

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

Semi-Heyting algebras were introduced by the second-named author during 1983-85 as an abstraction of Heyting algebras. The first results on these algebras, however, were published only in 2008 (see [San08]). Three years later, in [San11], he initiated the investigations into the variety DHMSH of dually hemimorphic semi-Heyting algebras obtained by expanding semi-Heyting algebras with a dually hemimorphic operation. His investigations were continued in a series of papers thereafter. He also had raised the problem of finding logics corresponding to subvarieties of DHMSH, such as the variety DMSH of De Morgan semi-Heyting algebras, and DPCSH of dually pseudocomplemented semi-Heyting algebras, as well as logics to 2, 3, and 4-valued DHMSH-matrices. In this paper, we first present a Hilbert-style axiomatization of a new implicative logic called--Dually hemimorphic semi-Heyting logic, (DHMSH, for short)-- as an expansion of semi-intuitionistic logic by the dual hemimorphism as the negation and prove that it is complete with respect to the variety DHMSH of dually hemimorphic semi-Heyting algebras as its equivalent algebraic semantics (in the sense of Abstract Algebraic Logic). Secondly, we characterize the (axiomatic) extensions of DHMSH in which the Deduction Theorem holds. Thirdly, we present several logics, extending the logic DHMSH, corresponding to several important subvarieties of the variety DHMSH, thus solving the problem mentioned earlier. We also provide new axiomatizations for Moisil's logic and the 3-valued Lukasiewicz logic.Comment: 66 pages, 3 figure