Graded congruences on leavitt path algebras over the boolean semifield and applications

In this article, we introduce and characterize graded congruences on a hemiring graded by an arbitrary group. Then we give a structure theorem for graded ideals of the Leavitt path algebra of a row-finite directed graph [Formula: see text] over an arbitrary semifield, as well as provide a structure theorem for graded congruences of the Leavitt path algebra [Formula: see text] of [Formula: see text] over the Boolean semifield [Formula: see text]. Consequently, we obtain that the lattice of all graded congruences of the Leavitt path algebra [Formula: see text] is isomorphic to both the lattice of all graded ideals of the Leavitt path algebra [Formula: see text] and the lattice of all saturated hereditary subsets of [Formula: see text]. We also show that this is, in general, not true for Leavitt path algebras over semifields. Moreover, we establish that the partially ordered set of all graded congruences of the Leavitt path algebra [Formula: see text] is a retract of the partially ordered set of all congruences of the graph inverse semigroup of [Formula: see text], and give graph-theoretic conditions for these two sets to be lattice-isomorphic.