ACTA issues

Linear representations of regular rings and complemented modular lattices with involution

Christian Herrmann, Marina Semenova

Acta Sci. Math. (Szeged) 82:3-4(2016), 395-442

Abstract. Faithful representations of regular $\ast $-rings and modular complemented lattices with involution within orthosymmetric sesquilinear spaces are studied within the framework of Universal Algebra. In particular, the correspondence between classes of spaces and classes of representable structures is analyzed; for a class $\mathcal{S}$ of spaces which is closed under ultraproducts and non-degenerate finite-dimensional subspaces, the class of representable structures is shown to be closed under complemented [regular] subalgebras, homomorphic images, and ultraproducts. Moreover, this class is generated by its members which are isomorphic to subspace lattices with involution [endomorphism $\ast $-rings, respectively] of finite-dimensional spaces from $\mathcal{S}$. Under natural restrictions, this result is refined to a $1$-$1$-correspondence between the two types of classes.

DOI: 10.14232/actasm-015-283-5

AMS Subject Classification (1991): 06C20, 16E50, 16W10, 51D25

Keyword(s): sesquilinear space, endomorphism ring, regular ring with involution, lattice of subspaces, complemented modular lattice with involution, representation, semivariety, variety

Received May 8, 2015, and in revised form May 22, 2016. (Registered under 33/2015.)