Lattice Uniformities Generated by Filters

On Lattices of Uniformities

“…This was generalized in [4] for sectionally complemented lattices. By By [9,Theorem VI.4.9] for any lattice L the system of all congruence relations is a Brouwerian lattice.…”

“…In general, LUD(L) is not modular; this doesn't contradict 3.6 since in general LUD(L) is not a sublattice of LU(L) (see [4, p. 518]). If L is sectionally complemented, then LUD(L) = LU(L) (see [4,Proposition 2.13]). …”

“…3 An analogous result for ring topologies on a ring with unit is contained in [25,Satz 2.2]. 4 Basile and Traynor [7] used the term "monotonely Cauchy" instead of "locally exhaustive". Proposition 5.11 we show that for a lattice uniformity w on a Boolean algebra L, the center of LU(L, w) := {u ∈ LU(L) : u ≤ w} is closed iff w is exhaustive.…”

“…For this, we also study the relationship between complements in L and complements in {u|[a, b ] : u ∈ L} for a, b ∈ L with a ≤ b (Corollary 5.7). Combining the general result 5.7 with the result of [4] saying that all exhaustive lattice uniformities on a modular complemented lattice form a Boolean algebra (Theorem 5.1 (a)), we can determine the center of the lattice of all locally exhaustive uniformities on a relative complemented lattice (Corollary 5.8). Similarly, combining 5.7 with the theorem of Avallone and Vitolo [2] saying that all exhaustive D-uniformities on a D-lattice form a Boolean algebra (Theorem 5.1 (b)), we determine the center of the lattice of all locally exhaustive D-uniformities on a generalized D-lattice (Corollary 5.9).…”

Modular functions on multilattices

“…This result allowed to use the theory of lattice uniformities developed in [22], [23], [27], [7], to extend to modular functions on lattices many results of classical measure theory, which have applications in particular in non-commutative measure theory and in fuzzy measure theory (see, for example, [1], [2], [4], [5], [6], [3], [8], [9], [13], [14], [15], [21], [23], [24], [25], [27]). …”