On the LCM Lattice of Powers of Monomial Ideal in Three Variables

Abstract

Commutative algebra, graph theory, algebraic geometry and combinatorics, semigroup rings and combinatorial optimization, integer programming, and other fields are all related with monomial algebra. Suppose K is a field, S = K[x1, ..., xn] is a polynomial ring with n number of variables, lcm(J), the LCM lattice of J, may be constructed for each mono- mial ideal J ⊂ S = K[x1, ..., xn]. We categorise the powers of monomial ideals in terms of their qualities in terms of LCM lattices. The lcm lattice of the ideal J created by a set of monomials is defined as a lattice with every member being a least common multiple of generating set of J. In particular, we discuss about powers of monomial ideal and staircase of monomial ideal in three variable. The powers and products of monomial ideals in poly- nomial rings are the subject of this thesis. On the polynomial ring K[x1, x2, x3], we derive conditions for the power of monomial ideals. The powers of monomial ideals are easier to calculate when the requirements are met.