Basic algebras, logics, trends and applications

2015 ◽  
Vol 08 (03) ◽  
pp. 1550040 ◽  
Author(s):  
Ivan Chajda

The classical logic was axiomatized algebraically by means of Boolean algebras in 19th century by George Boole. Similar attempts went on 20th century for algebraic axiomatization of non-classical logics, e.g. intuitionistic logics (Brouwer and Heyting algebras), many-valued logics (Łukasiewicz, Chang’s MV-algebras, Post algebras), the logic of quantum mechanics (orthomodular lattices and posets) and fuzzy logics (residuated lattices). In this paper, we are focused in a common generalization of MV-algebras and orthomodular lattices. The resulting algebras, called basic algebras, have surprisingly strong and interesting properties and they can be investigated in their own. The aim of the paper is to get an overview of results reached during the last decade.

2019 ◽  
Vol 69 (3) ◽  
pp. 533-540
Author(s):  
Ivan Chajda ◽  
Helmut Länger

Abstract Basic algebras were introduced by Chajda, Halaš and Kühr as a common generalization of MV-algebras and orthomodular lattices, i.e. algebras used for formalization of non-classical logics, in particular the logic of quantum mechanics. These algebras were represented by means of lattices with section involutions. On the other hand, classical logic was formalized by means of Boolean algebras which can be converted into Boolean rings. A natural question arises if a similar representation exists also for basic algebras. Several attempts were already realized by the authors, see the references. Now we show that if a basic algebra is commutative then there exists a representation via certain semirings with involution similarly as it was done for MV-algebras by Belluce, Di Nola and Ferraioli. These so-called basic semirings, their ideals and congruences are studied in the paper.


Studia Logica ◽  
2021 ◽  
Author(s):  
D. Fazio ◽  
A. Ledda ◽  
F. Paoli

AbstractThe variety of (pointed) residuated lattices includes a vast proportion of the classes of algebras that are relevant for algebraic logic, e.g., $$\ell $$ ℓ -groups, Heyting algebras, MV-algebras, or De Morgan monoids. Among the outliers, one counts orthomodular lattices and other varieties of quantum algebras. We suggest a common framework—pointed left-residuated $$\ell $$ ℓ -groupoids—where residuated structures and quantum structures can all be accommodated. We investigate the lattice of subvarieties of pointed left-residuated $$\ell $$ ℓ -groupoids, their ideals, and develop a theory of left nuclei. Finally, we extend some parts of the theory of join-completions of residuated $$\ell $$ ℓ -groupoids to the left-residuated case, giving a new proof of MacLaren’s theorem for orthomodular lattices.


2010 ◽  
Vol 60 (6) ◽  
Author(s):  
Jiří Rachůnek ◽  
Dana Šalounová

AbstractBounded Rℓ-monoids form a large subclass of the class of residuated lattices which contains certain of algebras of fuzzy and intuitionistic logics, such as GMV-algebras (= pseudo-MV-algebras), pseudo-BL-algebras and Heyting algebras. Moreover, GMV-algebras and pseudo-BL-algebras can be recognized as special kinds of pseudo-MV-effect algebras and pseudo-weak MV-effect algebras, i.e., as algebras of some quantum logics. In the paper, bipartite, local and perfect Rℓ-monoids are investigated and it is shown that every good perfect Rℓ-monoid has a state (= an analogue of probability measure).


1970 ◽  
pp. 47-55
Author(s):  
Sarah Limorté

Levantine immigration to Chile started during the last quarter of the 19th century. This immigration, almost exclusively male at the outset, changed at the beginning of the 20th century when women started following their fathers, brothers, and husbands to the New World. Defining the role and status of the Arab woman within her community in Chile has never before been tackled in a detailed study. This article attempts to broach the subject by looking at Arabic newspapers published in Chile between 1912 and the end of the 1920s. A thematic analysis of articles dealing with the question of women or written by women, appearing in publications such as Al-Murshid, Asch-Schabibat, Al-Watan, and Oriente, will be discussed.


2014 ◽  
Vol 3 (3) ◽  
pp. 114-143
Author(s):  
Gabriel Crumpei ◽  
Maricel Agop ◽  
Alina Gavriluţ ◽  
Irina Crumpei

Abstract In this paper, we aim at an exercise that is transdisciplinary, involving science and religion, and interdisciplinary, involving disciplines and theories which appeared in the second half of the 20th century (e.g., topology, chaos theory, fractal geometry, non-linear dynamics, all of which can be found in the theory of complex systems). The latter required the reformulation of quantum mechanics theories starting with the beginning of the century, based on the substance-energy-information triangle. We focus on information and we also attempt a transdisciplinary approach to the imaginary from a psychological - physical - mathematical perspective, but the religious perspectives find their place along with the philosophical or even philological vision


1969 ◽  
Vol 21 ◽  
pp. 1-368 ◽  
Author(s):  
Anthony K. Higgins

The first recorded landing by Europeans on the coast of northern East Greenland (north of 69°N) was that of William Scoresby Jr., a British whaler, in 1822. This volume includes a chronological summary of the pioneer 19th century exploration voyages made by British, Danish, Norwegian, Swedish, French and German expeditions – all of whom reported that the region had previously been occupied by the Inuit or Eskimo; also included are brief outlines of the increasing number of government and privately sponsored expeditions throughout the 20th century, whose objectives included cartography, geology, zoology, botany, trapping and the ascent of the highest mountain summits. In 1934 the Place Name Committee for Greenland was established, the tasks of which included a review of all place names hitherto recorded on published maps of Greenland, their formal adoption in danicised form, and the approval or rejection of new name proposals. In northern East Greenland, by far the largest numbers of new place names were those proposed by scientists associated with Lauge Koch's geological expeditions that lasted from 1926 until 1958. This volume records the location and origin of more than 3000 officially approved place names as well as about 2650 unapproved names. The author's interest in the exploration history and place names of northern East Greenland started in 1968, when the Geological Survey of Greenland initiated a major five-year geological mapping programme in the Scoresby Sund region. Systematic compilation of names began about 1970, initially with the names given by William Scoresby Jr., and subsequently broadened in scope to include the names proposed by all expeditions to northern East Greenland. The author has participated in 16 summer mapping expeditions with the Survey to northern East Greenland. Publication of this volume represents the culmination of a lifetime working in the Arctic.


1999 ◽  
Vol 26 (1) ◽  
pp. 27-51
Author(s):  
Jan Richard Heier

Accounting has always been utilitarian in nature. It adapts to the changes in the business environment by meeting the need for new types of information. The change in waterborne transportation in the U.S. during the 19th century provides an example of such an environmental change that led to a need for accounting adaptation. With the advent of the steamboat, old accounting methods were modified and new ones created to meet the changes in the business environment. In the process, a standardized ships-accounting model was developed. The model can be seen in the accounting records of three ships that sailed at the beginning of the 20th century.


2016 ◽  
Vol 3 (2) ◽  
pp. 3 ◽  
Author(s):  
Mara Calvini ◽  
Maria Stella Siori ◽  
Spartaco Gippoliti ◽  
Marco Pavia

The revised catalogue of primatological material stored in the Museo Regionale di Scienze Naturali of Torino and in the Dipartimento di Scienze della Vita e Biologia dei Sistemi of the Università degli Studi di Torino and belonging to the historical material of the Torino University is introduced. The material, 494 specimens belonging to 399 individuals of 122 taxa, is of particular importance since specimens were mainly obtained during the 19th Century and the beginning of the 20th Century. A relevant part of the collection was created by the collaborators of the Museum, among which it is worth to mention F. De Filippi, A. Borelli and E. Festa, while other material came from purchases and donations from private people or the Royal Zoological Garden of Torino. Great part of the specimens is stuffed but also the osteological materials are of particular importance, as many of them derived from the specimens before being prepared and consisting of skulls or more or less complete skeletons. After this revision, the Lectotype and Paralectotypes of <em>Alouatta</em> <em>palliata</em> <em>aequatorialis</em> have been selected, and the type-specimen of the <em>brunnea</em> variety of <em>Cebus</em> <em>albifrons</em> <em>cuscinus</em> has been recognized. In addition, some specimens of particular historical-scientific importance have also been identified and here presented for the first time.


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