Linear and multilinear functional identities in a prime ring with applications

This paper approaches some universal-algebraic properties of the two kinds of multilinear functions [Formula: see text] and [Formula: see text] in a prime ring [Formula: see text], where [Formula: see text] are variable elements, [Formula: see text]. We shall demonstrate an algebraic procedure of deriving necessary and sufficient conditions for the two multilinear functional identities [Formula: see text] and [Formula: see text] to hold for all [Formula: see text], [Formula: see text]. Subsequently, we use these multilinear functional identities to describe the invariance properties of the products [Formula: see text] [Formula: see text], [Formula: see text], [Formula: see text] with respect to the eight commonly-used types of generalized inverses of two MP-invertible elements [Formula: see text] and [Formula: see text] in a prime ring [Formula: see text] with an identity element 1 and ∗-involution.

Is mainstream LCA linear?

Purpose It is frequently mentioned in literature that LCA is linear, without a proof, or even without a clear definition of the criterion for linearity. Here we study the meaning of the term linear, and in relation to that, the question if LCA is indeed linear. Methods We explore the different meanings of the term linearity in the context of mathematical models. This leads to a distinction between linear functions, homogeneous functions, homogenous linear functions, bilinear functions, and multilinear functions. Each of them is defined in accessible terms and illustrated with examples. Results We analyze traditional, matrix-based, LCA, and conclude that LCA is not linear in any of the senses defined. Discussion and conclusions Despite the negative answer to the research question, there are many respects in which LCA can be regarded to be, at least to some extent, linear. We discuss a few of such cases. We also discuss a few practical implications for practitioners of LCA and for developers of new methods for LCI and LCIA.