A study of the quasi covering dimension of finite lattices

Abstract

The notion of dimension for posets has been studied extensively (see, for example, Bae in J Korean Math Soc 36(1):15–36, 1999, Dube et al. in Discr Math 338:1096–1110, 2015, Gierz et al. in A compendium of continuous lattices. Springer, Berlin, 1980, Trotter in Combinatorics and partially ordered sets: dimension theory. Johns Hopkins University Press, Baltimore, 1992, Vinokurov in Soviet Math Dokl 168(3):663–666, 1966, Zhang et al. in Discr Math 340(5):1086–1091, 2017). Moreover, in the view of matrix theory, some of these dimensions, such as the order dimension, the Krull dimension and the covering dimension, have been studied using the so-called order and incidence matrices (see Boyadzhiev et al. in Appl Math Comput 333:276–285, 2018, Dube et al. in Filomat 31(10):2901–2915, 2017, Georgiou et al. in Quaest Math 39(6):797–814, 2016). In this paper, we define a new dimension for finite lattices, called the quasi covering dimension, and we study many of its properties. We characterize it using matrices and we present an algorithm for computing this dimension.