scholarly journals Sectionally Pseudocomplemented Residual Lattice

Author(s):  
MD Zaidur Rahman ◽  
Md Abdul Kalam Azad ◽  
Md Nazmul Hasan

At first, we recall the basic concept, By a residual lattice is meant an algebra L = (L,∨,∧,∗,o,0,1) such that (i) L = (L,∨,∧,0,1) is a bounded lattice, (ii) L = (L,∗,1) is a commutative monoid, (iii) it satisfies the so-called adjoin ness property: (x ∨ y) ∗ z = y if and only if y ≤ z ≤ x o y Let us note [7] that x ∨ y is the greatest element of the set (x ∨ y) ∗ z = y Moreover, if we consider x ∗ y = x ∧ y , then x o y is the relative pseudo-complement of x with respect to y, i. e., for ∗ = ∧ residuated lattices are just relatively pseudo-complemented lattices. The identities characterizing sectionally pseudocomplemented lattices are presented in [3] i.e. the class of these lattices is a variety in the signature {∨,∧,o,1}. We are going to apply a similar approach for the adjointness property: Key words: Residuated lattice; non Distributive; Residuated Abeliean; commutative monoid: DOI: http://dx.doi.org/10.3329/diujst.v6i2.9345 DIUJST 2011; 6(2): 53-54

2020 ◽  
Vol 70 (2) ◽  
pp. 239-250
Author(s):  
Ivan Chajda ◽  
Jan Kühr ◽  
Helmut Länger

Abstract It is known that every relatively pseudocomplemented lattice is residuated and, moreover, it is distributive. Unfortunately, non-distributive lattices with a unary operation satisfying properties similar to relative pseudocomplementation cannot be converted in residuated ones. The aim of our paper is to introduce a more general concept of a relatively residuated lattice in such a way that also non-modular sectionally pseudocomplemented lattices are included. We derive several properties of relatively residuated lattices which are similar to those known for residuated ones and extend our results to posets.


Filomat ◽  
2020 ◽  
Vol 34 (4) ◽  
pp. 1223-1239
Author(s):  
Saeed Rasouli

The paper is devoted to introduce the notions of some types of stabilizers in non-commutative residuated lattices and to investigate their properties. We establish a connection between (contravariant) Galois connection and stabilizers of a residuated lattices. If A is a residuated lattice and F be a filter of A, we show that the set of all stabilizers relative to F of a same type forms a complete lattice. Furthermore, we prove that ST - F?l, ST - Fl and ST - Fs are pseudocomplemented lattices.


Author(s):  
А. Кошбаев

Аннотация: Бул макалада санат-насыят ырларды жаштарга тарбия таалим берүүдө, адам болуп калыптануусуна өзгөчө орунга ээ. Акындык жанрда ыр түрүндө, комуздун коштоосунда же жөн гана ооз эки айтыш менен айтылган. Санат-насыят, терме мактоо ырлары менен жаштардан баштап улгайганга чейин акыл туюмун өстүрүп келген. Акындар эл аралап санат-насыят ырларын ырдашкан. Эл топтолгон тойлордо, жыйындарда тарбиялык мааниси бар ырларды көпчүлүккө жайылткан. Бул тарбиялык мааниси бар ырларды көпчүлүк өздөрүнүн балдарына тарбия берүүдө колдонушкан жана кулактарына сиңиришкен. Токтогулдун «Өмүр», «Карылык», «Насыят», «Санат», «Үлгү ырлар», «Нускалуу ырлар», «Терме», «Курдаштын көөнүн билип өт» деген философиялык ойлорго бай, педагогикалык маңызы терең ырлары жөнүндө автор баяндайт. Токтогулдун чыгармачылыгындагы эң негизги концепция-адам эң жогорку турган кымбат нерсе, улуу идеал. Акындын өзүнүн адамды асыл зат катары жогору баалашы, ага өтө гумандуулук менен мамиле кылгандыгы улуулугу болуп саналары жөнүндө сңз болот. Түйүндүү сөздөр: багыттоочу, гумандуулук, насыят, педагогикалык маңызы, санат, тарбиялоочу, терме, философиялык ойлор, элдик. Аннотация: Назидательные и нравоучительные народные песни занимают особое место в деле воспитания молодежи и становлении их как личности. Песни исполнялись акынами как в сопровождении комуза, так и без него. В данной статье показано воспитательное значение нравоучительных песен Т. Сатылганова. Т. Сатылганов через свои песни, которые несли воспитательные, созидательные функции, передавал гуманистические идеи, распространял передовые мысли среди народных масс. Автор говорит о том, что песни Токтогула «Өмүр», «Карылык», «Насыят», «Санат», «Үлгү ырлар», «Нускалуу ырлар», «Терме», «Курдаштын көөнүн билип өт» полны философских размышлений, педагогического содержания. Он раскрывает основную концепцию творчества Токтогула о том, что человек наивысшее, самое ценное существо, великий идеал. Так же автор подчеркивает, что слова песен Токтогула отличаются образностью, обладают эмоциональной силой и возвышенностью. Акын меткими и точными словами дает характеристику как человеку, так и явлениям жизни. Каждый слушатель или читатель получает для себя из песен Токтогула жизненный опыт и делает выводы. Ключевые слова: направляющий, гуманизм, назидание, педагогическое содержание, поучение, воспитательный, сборный, философские мысли, народный. Abstract: Edifying and moralizing folk songs occupy a special place in the education of young people and their formation as a person. The songs were performed by akyns both accompanied by komuz and without it. This article shows the educational value of the moralizing songs of T. Satylganov. T. Satylganov through his songs, which carried educational, creative functions, transmitted humanistic ideas, spread advanced thoughts among the masses. The author says that the songs of the Toktogul "Omur", "Karylyk", "Naziat", "Sanat", "Ulgu yrlar", "Nuscaluu, yrlar", "Terme", "Kurdashtyn konun bilip ot" complete philosophical reflection, and pedagogical content. He reveals the basic concept of creativity Toktogul that man is the highest, most valuable creature, the great ideal. The author also emphasizes that the words of the songs of Toktogul differ in imagery, have emotional power and sublimity. Akyn apt and precise words gives a description of both people and phenomena of life. Each listener or reader gets from the songs of Toktogul's life experiences and draws conclusions. Key words: guide, humanism edification, teaching content, teaching educational, general, philosophy, folk.


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).


2021 ◽  
Author(s):  
Masoud Haveshki

Abstract We define the essential extension of a filter in the residuated lattice A associated to an ideal of L(A) and investigate its related properties. We prove the residuated lattice A is a Boolean algebra, G(RL)-algebra or MV -algebra if and only if the essential extension of {1} associated to A \ P is a Boolean filter, G-filter or MV -filter (for all P ∈ SpecA), respectively. Also, some properties of lattice of essential extensions are studied.


2019 ◽  
Vol 24 (2) ◽  
pp. 723-729
Author(s):  
Ivan Chajda ◽  
Helmut Länger

Abstract In a previous paper, the authors defined two binary term operations in orthomodular lattices such that an orthomodular lattice can be organized by means of them into a left residuated lattice. It is a natural question if these operations serve in this way also for more general lattices than the orthomodular ones. In our present paper, we involve two conditions formulated as simple identities in two variables under which this is really the case. Hence, we obtain a variety of lattices with a unary operation which contains exactly those lattices with a unary operation which can be converted into a left residuated lattice by use of the above mentioned operations. It turns out that every lattice in this variety is in fact a bounded one and the unary operation is a complementation. Finally, we use a similar technique by using simpler terms and identities motivated by Boolean algebras.


Author(s):  
ANŽE RODE

Povzetek Za zagotavljanje varnosti mora Slovenija v prihodnji strategiji upoštevati tudi silo, še posebej pri odvračanju groženj. Upoštevajoč omejene vire in spremenljivo, izzivov polno okolje, je težko najti celovito strategijo, ki bi omogočala varnost, napredek in ekonomski razvoj Slovenije. Rešitev vidimo v odvračanju, tako z zavezniki, kot z lastnimi silami. Da bi bili verodostojen partner v Zavezništvu, moramo prevzeti svoj delež bremena in razviti lastne, z zavezniki dogovorjene zmogljivosti. Z vojaške perspektive so razvidni trije ključni elementi sprejemljive strategije: močno zavezništvo, razvite lastne vojaške zmogljivosti in odpornost družbe. Pri razvoju zmogljivosti, mora Slovenska vojska upoštevati dogovorjeno metodologijo. Ključnega pomena za uspešen razvoj zmogljivosti in bojevanje v “večdimenzionalni bitki” je poveljevanje s poslanstvom. Ključne besede: Slovenska vojska, Odvračanje, Vojaške zmogljivosti, Strategija, Poveljevanje s poslanstvom Abstract Slovenia will use coercion – deterrence by denial in particular – as a basic concept for Slovenia’s future strategy. Taking into account scare resources, and a dynamic and volatile environment, it is difficult to find a wholly adequate strategy to provide for the safety, progress, and economic development of Slovenians. But deterrence – created by both our own capabilities and those gained through NATO membership – is our best way forward. In order to be reliable partner in NATO Slovenia has to take its share of the burden, and develop its own capabilities. From a military perspective, there are three key elements for successful strategy: strong alliances, well-developed national military capabilities, and the resilience of society. In developing its military capabilities the Slovenian Armed Forces (SAF) has to apply a DOTMLPF framework. Here we see mission command pivotal for success in development, and in Multi Domain Battle. Key words: Slovenian Armed Forces, Deterrence, Military Capabilities, Strategy, Mission command


2018 ◽  
Vol 52 (1) ◽  
pp. 23-42 ◽  
Author(s):  
Maryam Ghorani

In this paper, an association is organized between the theory of tree automata on one hand and the hyperstructures on the other hand, over complete residuated lattices. To this end, the concept of order of the states of a complete residuated lattice-valued tree automaton (simply L-valued tree automaton) is introduced along with several equivalence relations in the set of the states of an L-valued tree automaton. We obtain two main results from this study: one of the relations can lead to the creation of Kleene’s theorem for L-valued tree automata, and the other one leads to the creation of a minimal v-valued tree automaton that accepts the same language as the given one.


2021 ◽  
pp. 1-11
Author(s):  
Yan Yan Dong ◽  
Jun Tao Wang

In this paper, we first point out some mistakes in [12]. Especially the Theorem 3.9 [12] showed that: Let A be residuated lattice and ∅ ≠ X ⊆ A, then the least ideal containing X can be expressed as: 〈X〉 = {a ∈ A|a ≤ (·· · ((x1 ⊕ x2) ⊕ x3) ⊕ ·· ·) ⊕ xn, xi ∈ X, i = 1, 2 ·· · , n}. But we present an example to illustrate the ideal generation formula may not hold on residuated lattices. Further we give the correct ideal generation formula on residuated lattices. Moreover, we extend the concepts of annihilators and α-ideals to MTL-algebras and focus on studying the relations between them. Furthermore, we show that the set Iα (M) of all α-ideals on a linear MTL-algebra M only contains two trivial α-ideals {0} and M. However, the authors [24] studied the structure of Iα (M) in a linear BL-algebra M, which means some results with respect to Iα (M) given in [24] are trivial. Unlike that, we investigate the lattice structure of Iα (M) on general MTL-algebras.


2011 ◽  
Vol 07 (03) ◽  
pp. 499-513 ◽  
Author(s):  
SHOKOOFEH GHORBANI

In this paper, the concept of intuitionistic fuzzy sets is applied to residuated lattices. The notion of intuitionistic fuzzy filters of a residuated lattice is introduced and some related properties are investigated. The characterizations of intuitionistic fuzzy filters are obtained. We show that the set of all the intuitionistic fuzzy filters of a residuated lattice forms a complete lattice and we find the distributive sublattices of it. Finally, the correspondence theorem for intuitionistic fuzzy filters is established.


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