Abstract
On the basis of join-continuous semigroups \((L,\le,\,\&\,)\), this paper demonstrates among other things that on every complete ortholattice there exists an L-valued topology such that order convergence is L-topological (this is not always the case with respect to ordinary topologies). Further, the concept of lower semicontinuous lattice-valued maps permits an extension of the omega-functor \(\omega _{L}\) to the general setting of L-valued topological spaces where the importance of \( \omega_L\) lies in the replacement of ordinary topologies by L-valued topologies. It is shown that \(\omega _{L}\) has a right adjoint functor iff the underlying lattice \((L,\le)\) is continuous.
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Hohle, U., Kubiak, T. Many Valued Topologies and Lower Semicontinuity. Semigroup Forum 75, 1–17 (2007). https://doi.org/10.1007/s00233-006-0652-z
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DOI: https://doi.org/10.1007/s00233-006-0652-z