Graded simplicities of Leavitt path algebras with coefficients in Clifford semifield and k-semifield

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Using the natural Z -grading, Graded Simplicity and Graded Uniqueness Theorems have been obtained in the past for Leavitt path algebras over fields and commutative unital rings. In this paper we extend some of these results to the more general setting of Leavitt path algebras over Clifford semifield and k-semifield. Clifford semifield and k-semifield are natural generalizations of bounded distributive lattices as well as of fields. As evident from some previous works, Leavitt path algebra in these settings is not exactly the same as that in case of fields and commutative rings. Here we characterize graded full k-ideal simplicity of Leavitt path algebra over Clifford semifield and graded k-ideal simplicity over k-semifield by the graph theoretic properties of the underlying directed graph. We apply our simplicity theorems on the Leavitt path algebra of rose graph with one petal to establish that S [ x , x − 1 ] is graded full k-ideal simple (respectively, graded k-ideal simple) for any Clifford semifield (respectively, k-semifield) S. Finally we obtain versions of Graded Uniqueness Theorem for Leavitt path algebras with coefficients in Clifford semifield and k-semifield.