پالایه‌های بیشین، اول و اول کمین در مشبکه‌های مانده‌دار

رسولی, سعید; خاکسارحقانی, فرهاد

doi:10.22052/scj.2023.243375.1032

پالایه‌های بیشین، اول و اول کمین در مشبکه‌های مانده‌دار

نوع مقاله : مقاله پژوهشی

نویسندگان

¹ گروه ریاضی، دانشکده مهندسی سیستم‌های هوشمند و علوم داده، دانشگاه خلیج فارس، بوشهر، ایران

² گروه ریاضی، دانشکده علوم پایه، دانشگاه آزاد اسلامی واحد شهرکرد، شهرکرد، ایران

10.22052/scj.2023.243375.1032

چکیده

در این مقاله، به بررسی و مطالعه پالایه‌های بیشین، اول، و اول کمین در مشبکه‌های مانده‌دار می‌پردازیم و مثال‌هایی از مشبکه‌های مانده‌دار ارائه می‌دهیم. ابتدا به معرفی پالایه‌های یک مشبکه مانده‌دار می‌پردازیم و سپس نشان می‌دهیم هر پالایه سره مشمول در یک پالایه بیشین و در نتیجه مشمول در یک پالایه اول است. پالایه‌های اول در یک MTL-جبر را بازشناسی کرده و گزاره بنیادین پالایه‌های اول را برای مشبکه‌های مانده‌دار بیان و اثبات می‌کنیم. مشبکه‌های مانده‌داری را بازشناسی می‌کنیم که هر پالایه در آنها اصلی است و هم‌پوچسازها را با استفاده از پالایه‌های اول مورد بررسی قرار می‌دهیم. سپس، پالایه‌های اول کمین در یک مشبکه‌ مانده‌دار را معرفی کرده و گزاره بنیادین پالایه‌های اول کمین را بیان و اثبات می‌کنیم. در انتها، پالایه‌های بخش‌یاب در یک مشبکه مانده‌دار را معرفی کرده و پالایه‌های اول کمین را با استفاده از آنها مورد بازشناسی قرار می‌دهیم.

کلیدواژه‌ها

عنوان مقاله [English]

Maximal, Prime and Minimal prime filters in Residuated Lattices

نویسندگان [English]

Saeed Rasouli ¹
Farhad Khaksar Haghami ²

¹ Department of Mathematics, Faculty of Intelligent Systems Engineering and Data Science, Persian Gulf University, Bushehr, Iran

² Department of Mathematics, Faculty of Basic Sciences, Shahrekord Branch, Islamic Azad University, Shahrekord, Iran

چکیده [English]

In this paper, by considering the notions of maximal, prime, and minimal prime filters in residual lattices, we study their properties and some examples of residuated lattices are given. It is reminded first the notion of a filter of a residuated lattice and then we show that each proper filter is contained in a maximal filter and therefore in a prime filter. We investigate the prime filters in an MTL algebra and state and prove the fundamental theorem of the prime filters for residuated lattices. We investigate the residuated lattices in which each its filter is principal, and by using the prime filters, we inspect co-annihilators. Then the notion of a minimal prime filter in a residuated lattice is introduced and a fundamental theorem for minimal prime filters in the residuated lattice is stated and proved. Finally, we introduce the divisor filters in a residuated lattice and by using them; we investigate the minimal prime filters

کلیدواژه‌ها [English]

Residuated Lattices
Filtes
Maximal filters
Prime filters
Minimal prime filters
Divisor filters

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