# Congruences defined by languages and filters

Abstract. The usual right congruence ~L can be generalized in the following manner: x ~_{\cal{L},L} y iff {z | xz \in L iff yz \in \cal{L} }, where \cal{L} is a family of languages. It turns out to be useful when \cal{L} is a filter with an additional property. Furthermore semifilters are introduced and studied. It is also possible to define congruences by filters. Assuming the (right) congruences to have finite index yields a generalization of the regular sets.
helmut@gauss.cam.wits.ac.za,

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