Let X be a non-empty set, and let P(X) denote the semigroup of partial transformations on X. For a non-empty subset Y of X, define PT(X, Y ) = { alpha E P(X) (dom alpha n Y)alpha C Y }. The semigroup PT(X, Y ) generalizes P(X) and consists of all partial transformations on X that leave Y invariant. In this paper, we investigate the natural partial order on PT (X, Y ) and characterize its left-compatible, right-compatible, minimal, and maximal elements. The results obtained extend and unify several known properties of P(X).
This article discusses differential identities involving Jordan and Lie products that connect an MA-semiring to its prime ideals. The main objective is to investigate how these identities impose various properties on the quotient MA-semiring determined by a prime ideal. In particular, we establish Herstein's theorem for MA-semirings, which was previously proved for rings.
In this paper we define and study a new kind of dimension called, semisimple dimension, that measures how far a module is from being semisimple. Like other kinds of dimensions, this is an ordinal valued invariant. It is shown that a Noetherian module with semisimple dimension is an Artinian module. Moreover, we involve the theory of graphs within of such dimension achieving some applications for the edge ideal of a graph.
. A matroid port is a clutter determined by the circuits of a matroid that contain a fixed point. A monomial ideal associated to a (binary) matroid port can be characterized by using its minimal generators and its minimal prime ideals. In this note we point out this characterization and we demonstrate that any fullsupported squarefree monomial ideal is the intersection of finitely many ideals associated to matroid ports. In the binary case, the matroid port decomposition is related to the irredundant primary decomposition of the ideal.
Let V be an infinite-dimensional vector space over a field of characteristic not equal to 2. Given a nondegenerate quadratic form f on V, we consider the Clifford algebra Cl(V,f). Any orthogonal linear transformation of V extends to a Bogolyubov automorphism of Cl(V,f). We obtain necessary and sufficient conditions for a Bogolyubov automorphism to be inner.
