scholarly journals Absolutely Free Multialgebras

2021 ◽  
Author(s):  
Marcelo Esteban Coniglio ◽  
Guilherme Vicentin de Toledo
Keyword(s):  
Algebraic Logic ◽  
Forgetful Functor ◽  
Left Adjoint ◽  
Paraconsistent Logics ◽  
Mapping Property ◽  
Strong Basis ◽  
Generating Set ◽  
Sigma Algebras ◽  
Unique Homomorphism ◽  
Natural Way

In abstract algebraic logic, many systems, such as those paraconsistent logics taking inspiration from da Costa's hierarchy, are not algebraizable by even the broadest standard methodologies, as that of Blok and Pigozzi. However, these logics can be semantically characterized by means of non-deterministic algebraic structures such as Nmatrices, RNmatrices and swap structures. These structures are based on multialgebras, which generalize algebras by allowing the result of an operation to assume a non-empty set of values. This leads to an interest in exploring the foundations of multialgebras applied to the study of logic systems. It is well known from universal algebra that, for every signature \(\Sigma\), there exist algebras over \(\Sigma\) which are absolutely free, meaning that they do not satisfy any identities or, alternatively, satisfy the universal mapping property for the class of \(\Sigma\)-algebras. Furthermore, once we fix a cardinality of the generating set, they are, up to isomorphisms, unique, and equal to algebras of terms (or propositional formulas, in the context of logic). Equivalently, the forgetful functor, from the category of \(\Sigma\)-algebras to Set, has a left adjoint. This result does not extend to multialgebras. Not only multialgebras satisfying the universal mapping property do not exist, but the forgetful functor \(\mathcal{U}\), from the category of \(\Sigma\)-multialgebras to Set, does not have a left adjoint. In this paper we generalize, in a natural way, algebras of terms to multialgebras of terms, whose family of submultialgebras enjoys many properties of the former. One example is that, to every pair consisting of a function, from a submultialgebra of a multialgebra of terms to another multialgebra, and a collection of choices (which selects how a homomorphism approaches indeterminacies), there corresponds a unique homomorphism, what resembles the universal mapping property. Another example is that the multialgebras of terms are generated by a set that may be viewed as a strong basis, which we call the ground of the multialgebra. Submultialgebras of multialgebras of terms are what we call weakly free multialgebras. Finally, with these definitions at hand, we offer a simple proof that multialgebras with the universal mapping property for the class of all multialgebras do not exist and that \(\mathcal{U}\) does not have a left adjoint.

Copeland algebras

1972 ◽  
Vol 37 (4) ◽  
pp. 646-656 ◽  
Author(s):  
Daniel B. Demaree
Keyword(s):  
Predicate Logic ◽  
Algebraic Logic ◽  
Boolean Algebras ◽  
Primary Concern ◽  
Cylindric Algebras ◽  
First Order ◽  
Polyadic Algebras ◽  
Algebraic Setting ◽  
Finite Set ◽  
Natural Way

It is well known that the laws of logic governing the sentence connectives—“and”, “or”, “not”, etc.—can be expressed by means of equations in the theory of Boolean algebras. The task of providing a similar algebraic setting for the full first-order predicate logic is the primary concern of algebraic logicians. The best-known efforts in this direction are the polyadic algebras of Halmos (cf. [2]) and the cylindric algebras of Tarski (cf. [3]), both of which may be described as Boolean algebras with infinitely many additional operations. In particular, there is a primitive operator, cκ, corresponding to each quantification, ∃υκ. In this paper we explore a version of algebraic logic conceived by A. H. Copeland, Sr., and described in [1], which has this advantage: All operators are generated from a finite set of primitive operations.Following the theory of cylindric algebras, we introduce, in the natural way, the classes of Copeland set algebras (SCpA), representable Copeland algebras (RCpA), and Copeland algebras of formulas. Playing a central role in the discussion is the set, Γ, of all equations holding in every set algebra. The reason for this is that the operations in a set algebra reflect the notion of satisfaction of a formula in a model, and hence an equation expresses the fact that two formulas are satisfied by the same sequences of objects in the model. Thus to say that an equation holds in every set algebra is to assert that a certain pair of formulas are logically equivalent.

scholarly journals Adjunctions and braided objects

2014 ◽  
Vol 13 (06) ◽  
pp. 1450019 ◽  
Author(s):  
Alessandro Ardizzoni ◽  
Claudia Menini
Keyword(s):  
Monoidal Category ◽  
Forgetful Functor ◽  
Left Adjoint ◽  
Tensor Algebra ◽  
Adjoint Functors ◽  
Base Category ◽  
The One ◽  
Braided Bialgebras

In this paper, we investigate the categories of braided objects, algebras and bialgebras in a given monoidal category, some pairs of adjoint functors between them and their relations. In particular, we construct a braided primitive functor and its left adjoint, the braided tensor bialgebra functor, from the category of braided objects to the one of braided bialgebras. The latter is obtained by a specific elaborated construction introducing a braided tensor algebra functor as a left adjoint of the forgetful functor from the category of braided algebras to the one of braided objects. The behavior of these functors in the case when the base category is braided is also considered.

scholarly journals Functoriality of the Schmidt Construction

2021 ◽  
Author(s):  
Juan Climent Vidal ◽  
Enric Cosme Llópez
Keyword(s):  
Left Adjoint ◽  
Sigma Algebras ◽  
Recursion Theorem ◽  
Comma Category ◽  
Object Mapping ◽  
Initial Object

After proving, in a purely categorial way, that the inclusion functor InAlg(Σ) from Alg(Σ), the category of many-sorted Σ-algebras, to PAlg(Σ), the category of many-sorted partial Σ-algebras, has a left adjoint FΣ, the (absolutely) free completion functor, we recall, in connection with the functor FΣ, the generalized recursion theorem of Schmidt, which we will also call the Schmidt construction. Next we define a category Cmpl(Σ), of Σ-completions, and prove that FΣ, labeled with its domain category and the unit of the adjunction of which it is a part, is a weakly initial object in it. Following this we associate to an ordered pair (α,f), where α=(K,γ,α) is a morphism of Σ-completions from F=(C,F,η) to G=(D,G,ρ) and f a homomorphism in D from the partial Σ-algebra A to the partial Σ-algebra B, a homomorphism ΥαG,0(f):Schα(f)B. We then prove that there exists an endofunctor, ΥαG,0, of Mortw(D), the twisted morphism category of D, thus showing the naturalness of the previous construction. Afterwards we prove that, for every Σ-completion G=(D,G,ρ), there exists a functor ΥG from the comma category (Cmpl(Σ)↓G) to End(Mortw(D)), the category of endofunctors of Mortw(D), such that ΥG,0, the object mapping of ΥG, sends a morphism of Σ-completion in Cmpl(Σ) with codomain G, to the endofunctor ΥαG,0.

𝔪-Stone Lattices

1972 ◽  
Vol 24 (6) ◽  
pp. 1027-1032 ◽  
Author(s):  
B. A. Davey
Keyword(s):  
Distributive Lattice ◽  
Principal Ideal ◽  
Forgetful Functor ◽  
Left Adjoint ◽  
Bounded Distributive Lattice ◽  
Stone Lattice ◽  
Complemented Lattice

A Stone lattice is a distributive, pseudo-complemented lattice L such that a* V a** = 1, for all a in L; or equivalently, a bounded distributive lattice L in which, for all a in L, the annihilator a⊥ = {b ∊ L|a ∧ b = 0} is a principal ideal generated by an element of the centre of L, namely a*.Thus it is natural to define an 𝔪-Stone lattice to be a bounded distributive lattice L in which, for each subset A of cardinality less than or equal to m, the annihilator A⊥ = {b ∊ L|a ∧ b = 0, for all a ∊ A} is a principal ideal generated by an element of the centre of L.In this paper we characterize 𝔪-Stone lattices, and show, by considering the lattice of global sections of an appropriate sheaf, that any bounded distributive lattice can be embedded in an 𝔪-Stone lattice, the embedding being a left adjoint to the forgetful functor.

INEVITABILITY OF REVIVAL: PRO AND COUNTER ANTINATALISM

2019 ◽  
Vol 19 (1) ◽  
pp. 80-88
Author(s):  
Nikolay S. Savkin
Keyword(s):  
Social Relations ◽  
Systematic Approach ◽  
Human Life ◽  
Laws Of Nature ◽  
Arthur Schopenhauer ◽  
Expected Effect ◽  
Philosophy Of Life ◽  
Active Subject ◽  
Over Time ◽  
Natural Way

Introduction. Radical pessimism and militant anti-natalism of Arthur Schopenhauer and David Benathar create an optimistic philosophy of life, according to which life is not meaningless. It is given by nature in a natural way, and a person lives, studies, works, makes a career, achieves results, grows, develops. Being an active subject of his own social relations, a person does not refuse to continue the race, no matter what difficulties, misfortunes and sufferings would be experienced. Benathar convinces that all life is continuous suffering, and existence is constant dying. Therefore, it is better not to be born. Materials and Methods. As the main theoretical and methodological direction of research, the dialectical materialist and integrative approaches are used, the realization of which, in conjunction with the synergetic technique, provides a certain result: is convinced that the idea of anti-natalism is inadequate, the idea of giving up life. A systematic approach and a comprehensive assessment of the studied processes provide for the disclosure of the contradictory nature of anti-natalism. Results of the study are presented in the form of conclusions that human life is naturally given by nature itself. Instincts, needs, interests embodied in a person, stimulate to active actions, and he lives. But even if we finish off with all of humanity by agreement, then over time, according to the laws of nature and according to evolutionary theory, man will inevitably, objectively, and naturally reappear. Discussion and Conclusion. The expected effect of the idea of inevitability of rebirth can be the formation of an optimistic orientation of a significant part of the youth, the idea of continuing life and building happiness, development. As a social being, man is universal, and the awareness of this universality allows one to understand one’s purpose – continuous versatile development.

HUKUM ISLAM DAN HUBUNGANNYA DENGAN LUAR NEGERI PASCA REFORMAS

2015 ◽  
Vol 19 (1) ◽  
Author(s):  
Salma Salma
Keyword(s):  
Human Rights ◽  
Foreign Policy ◽  
International Law ◽  
Multidisciplinary Approach ◽  
Islamic Law ◽  
Analysis Method ◽  
Strong Basis ◽  
Global Issues ◽  
Protection Of Human Rights ◽  
Scientific Fields

The development of Islamic law studies in Indonesia is increasingly interesting to follow. the use of a multidisciplinary approach to Islamic sciences, making the science of Islamic law not only a normative-theological analysis but also integrated with many scientific fields both in the sciences and the humanities. Contemporary global issues require observers and Islamic law reviewers to seriously review Islamic law in depth, one of the global issues that is currently interesting and has become a topic of discussion among many is the issue of the protection of human rights. Human rights formulation in international law cannot be separated from the issue of foreign policy. This paper will conduct a theoretical study of how the concept of Islamic Law itself protects human rights and how it relates to its relationship with post-reform foreign policy. This paper uses a comparative study between legislation and texts (verses) both in the Koran and the hadith, a comparative-critical analysis method makes it easier for the author to find substance in terms of answering the problem statement in this study. The results or conclusions obtained are that human rights are a reflection of carrying out Islamic law in order to realize the nature of universal human benefit. Islam considers that human rights are in accordance with sharia principles, namely protecting one's right to life. This is a strong basis for the study of Islamic law in contributing to the development of human rights principles in the international communityKeywords: Islamic Law, Human Rights, Globalization, International LawPerkembangan kajian hukum Islam di Indonesia makin menarik untuk diikuti. penggunaan pendekatan multidisipliner ilmu-ilmu keislaman, membuat ilmu hukum Islam tidak hanya bersifat normatif-teologis analisanya tapi sudah terintegrasi dengan banyak bidang keilmuan baik ilmu-ilmu sains maupun humaniora. Isu-isu global yang sifatnya kontemporer mengharuskan para pengamat dan pengkaji hukum Islam untuk serius melakukan telaah ulang terhadap ilmu hukum Islam secara mendalam, salah satu isu global yang saat ini menarik dan menjadi perbincangan banyak kalangan adalah soal perlindungan hak asasi manusia. Rumusan HAM dalam hukum internasional tidak bisa dilepaskan dengan persoalan politik luar negeri. Tulisan ini akan melakukan kajian teoritik tentang bagaimana konsep Hukum Islam itu sendiri terhadap perlindungan hak asasi manusia dan bagaimana pula terkait hubungannya dengan politik luar negeri pasca reformasi. Tulisan ini menggunakan studi komparatif antara perundangundangan dengan teks (ayat) baik itu di dalam Al-Quran maupun hadits, metode analisis-kritis komparatif memudahkan penulis menemukan substansi dalam hal untuk menjawab rumusan masalah dalam penelitian ini. Hasil atau kesimpulan yang didapat adalah HAM adalah refleksi untuk menjalankan syariat Islam demi mewujudkan hakikat kemaslahatan manusia secara universal. Islam memandang bahwa HAM sesuai dengan prinsip-prinsip syariah yakni melindungi hak hidup seseorang. Hal ini merupakan dasar yang kuat untuk kajian hukum Islam dalam memberikan kontribusi pada perkembangan prinsip-prinsip hak asasi manusia di dalam masyarakat internasional.Kata Kunci: Hukum Islam, Hak Asasi Manusia, Globalisasi, Hukum Internasional

Novel 1,2,3-Triazole-functionalized pyrido[3',2':4,5]furo[3,2-d]pyrimidin- 4(3H)-one Derivatives: Synthesis, Anticancer Activity, CoMFA and CoMSIA Studies

2020 ◽  
Vol 17 (12) ◽  
pp. 969-978
Author(s):  
Balakishan Vadla ◽  
Sailu Betala
Keyword(s):  
Anticancer Activity ◽  
Cell Line ◽  
Normal Cell ◽  
Rational Design ◽  
Human Cancer ◽  
Strong Basis ◽  
Hek 293 ◽  
Normal Cell Line ◽  
Human Cancer Cell Lines ◽  
Mcf 7

A series of novel triazole functionalized pyrido [3',2':4,5] furo[3,2-d] pyrimidin-4 (3H)-one derivatives 7a-p were prepared from ethyl furo[2,3-b]pyridine-2-carboxylate 3 on reaction with ammonia to afford furo[2,3-b]pyridine-2-carboxamide 4. This compound, on reaction with triethyl orthoformate TEOF, gave compound 5. Compound 5 on propargylation, followed by a reaction with substituted aryl azides under Sharpless reaction conditions, furnished triazole tagged pyrido [3',2':4,5]furo[3,2-d] pyrimidin-4(3H)-one derivatives. All the products 7a-p were screened against four human cancer cell lines, such as HeLa - Cervical cancer (CCL-2), COLO 205- Colon cancer (CCL-222), HepG2- Liver cancer (HB-8065), and MCF7 - Breast cancer (HTB-22) and one normal cell line (HEK 293). Compounds 7b, 7n, 7o and 7p, which showed promising anticancer activity, were identified and found to be non-toxic to normal cell line. Studies for HeLa, COLO205, HepG2, and MCF-7 using CoMFA and CoMSIA were carried out . Models from 3D-QSAR provided a strong basis for future rational design of more active and selective HeLa, COLO205, HepG2, and MCF-7 cell line inhibitors.

Duality Myths

2018 ◽  
Author(s):  
Elizabeth Schechter
Keyword(s):  
Narrative Art ◽  
Split Brain ◽  
Dramatic Form ◽  
Natural Way

This chapter addresses the intuitive fascination of the split-brain phenomenon. According to what I call the standard explanation, it is because we ordinarily assume that people are psychologically unified, while split-brain subjects are not psychologically unified, which suggests that we might not be unified either. I offer a different interpretation. One natural way of grappling with people’s failures to conform to various assumptions we make about them is to conceptualize them as having multiple minds. Such multiple-minds models take their most dramatic form in narrative art as duality myths. The split-brain cases grip people in part because the subjects strike them as living embodiments of such myths.

scholarly journals Residuated Structures and Orthomodular Lattices

Studia Logica ◽  
2021 ◽  
Author(s):  
D. Fazio ◽  
A. Ledda ◽  
F. Paoli
Keyword(s):  
Algebraic Logic ◽  
Residuated Lattices ◽  
Quantum Structures ◽  
Heyting Algebras ◽  
Quantum Algebras ◽  
Orthomodular Lattices ◽  
Lattice Of Subvarieties ◽  
Residuated Structures ◽  
Common Framework ◽  
Mv Algebras

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.

scholarly journals A generating set for the canonical ideal of HKG-curves

2021 ◽  
Vol 7 (1) ◽  
Author(s):  
Aristides Kontogeorgis ◽  
Ioannis Tsouknidas
Keyword(s):  
Generating Set ◽  
Canonical Ideal
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