Dual skew Heyting almost distributive lattices

Berhanu Assaye,Mihret Alamneh,Yeshiwas Mebrat,Lakshmi Narayan Mishra

doi:10.2478/amns.2019.1.00015

Abstract

Abstract In this paper, we introduce the concept of dual skew Heyting almost distributive lattices (dual skew HADLs) and characterise it in terms of dual HADL. We define an equivalence relation θ on a dual skew HADL L and prove that θ is a congruence relation on the equivalence class [x]θ so that each congruence class is a maximal rectangular subalgebra and the quotient [y]θ/θ is a maximal lattice image of [x]θ for any y ∈ [x]θ. Moreover, we show that if the set PI (L) of all the principal ideals of an ADL L with 0 is a dual skew Heyting algebra then L becomes a dual skew HADL. Further we present different conditions on which an ADL with 0 becomes a dual skew HADL.

Highlights

The foundation for the modern theory of skew lattice was laid by Jonathan Leech in 1989 [5]
G.C.Rao, Berhanu Assaye and M.V.Ratnamani in [3] introduced Heyting almost distributive lattice (ADL) (HADLs) as a generalisation of Heyting algebra in the class of ADLs, and they characterise an HADL in terms of the set of all its principal ideals
In the second section of this paper, we introduce the concept of dual skew HADLs

Summary

Introduction

The foundation for the modern theory of skew lattice was laid by Jonathan Leech in 1989 [5]. G.C.Rao, Berhanu Assaye and M.V.Ratnamani in [3] introduced Heyting ADLs (HADLs) as a generalisation of Heyting algebra in the class of ADLs, and they characterise an HADL in terms of the set of all its principal ideals. C. and Naveen Kumar K. introduced the concept of a dual Heyting ADL (dual HADL) [2] They derived a number of important laws and results satisfied by a dual HADL and dual L-ADL. In the second section of this paper, we introduce the concept of dual skew HADLs. We define an equivalence relation θ on a dual skew HADL and show that: θ is a congruence relation on each equivalence class, and each congruence class is a maximal rectangular subalgebra of the equivalence class and the quotient lattice [a]θ , where a ∈ [x]θ such that [x]θ is the equivalence class of L. Given that a skew ADL L with 0 on which the set of all the principal ideals of L is a dual skew Heyting algebra, we prove that L is a dual skew HADL

Preliminaries

Dual Skew HADLs