This paper studies localic products, traditional topological products, and L-topological products, and gives a com-
plete outline of the localic product. Comparisons of localic and L-topological products are absent in the literature,
and this paper answers longstanding open questions in that area as well as provides a complete proof of the classical
comparison theorem for localic and traditional topological products. This paper contributes several L-valued compar-
ison theorems, one of which states: the localic and L-topological products of L-topologies are order isomorphic if and
only if the localic product is L-spatial, providing L is itself spatial and the family of L-topological spaces is prime
separated". These last two conditions always hold in the traditional setting, capturing the traditional comparison
theorem as a special case, and the prime separation condition is satis¯ed by important lattice-valued examples that
include the fuzzy real line and the fuzzy unit interval for L any complete Boolean algebra, and the alternative fuzzy
real line and fuzzy unit interval for L any (semi)frame. Separation conditions help control the sloppy" behavior
of the L-topological product when jLj > 2, and several separation conditions are studied in this context; and it
should be noted that localic products have a point-free version of the product" separation condition considered in
this paper. The traditional comparison theorem is carefully proved both to ¯ll gaps in the extant literature and to
motivate the L-valued comparison theorem quoted above and reveal the special role played by cross sums of prime
(L-)open subsets. En route, characterizations are given of prime L-open subsets of certain L-products, which in turn
yield characterizations of prime open and irreducible closed subsets of traditional product spaces. |
