TY - JOUR

T1 - ON ESSENTIALITY AND IRREDUCIBILITY IN A LATTICE

AU - Sahoo, Tapatee

AU - Panackal, Harikrishnan

AU - Srinivas, Kedukodi Babushri

AU - Kuncham, Syam Prasad

N1 - Publisher Copyright:
© Palestine Polytechnic University-PPU 2022.

PY - 2022

Y1 - 2022

N2 - We consider a bounded lattice (L, ∧, ∨) with the smallest element 0 and the greatest element 1. In this paper, we deal with the essentiality concepts associated with a lattice. For an arbitrary element θ of L, we define a θ-e-irreducible element in L, which is an analogy to the concept of the e-irreducible submodule in a module over a ring. It is well known that e-irreducible submodules have no proper essential extension. Indeed, we prove this remains true for elements in a bounded lattice. We establish a relation between the θ-complement and θ-e-irreducible element with suitable examples. We define the notion θ-socle and prove several properties when a lattice is compactly generated. Further, we construct a generalized complement graph of a distributive lattice and relate the properties such as connectedness, diameter, and cut vertices to atoms in a lattice.

AB - We consider a bounded lattice (L, ∧, ∨) with the smallest element 0 and the greatest element 1. In this paper, we deal with the essentiality concepts associated with a lattice. For an arbitrary element θ of L, we define a θ-e-irreducible element in L, which is an analogy to the concept of the e-irreducible submodule in a module over a ring. It is well known that e-irreducible submodules have no proper essential extension. Indeed, we prove this remains true for elements in a bounded lattice. We establish a relation between the θ-complement and θ-e-irreducible element with suitable examples. We define the notion θ-socle and prove several properties when a lattice is compactly generated. Further, we construct a generalized complement graph of a distributive lattice and relate the properties such as connectedness, diameter, and cut vertices to atoms in a lattice.

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M3 - Article

AN - SCOPUS:85135227293

SN - 2219-5688

VL - 11

SP - 132

EP - 144

JO - Palestine Journal of Mathematics

JF - Palestine Journal of Mathematics

IS - Special Issue 3

ER -