(Skew) Filters in Residuated Skew Lattices

In this paper, we show the relationship between (skew) deductive system and (skew) filter in residuated skew lattices. It is shown that if a residuated skew lattice is conormal, then any skew deductive system is a skew filter under a condition and deductive system and skew deductive system are equivalent under some conditions too. It is investigated that in branchwise residuated skew lattice, filter, deductive system and skew deductive system are equivalent. We define some types of prime (skew) filters in residuated skew lattices and show the relationship between prime (skew) filters and residuated skew chains. It is proved that in prelinear residuated skew lattice any proper filter can be extended to a maximal, prime filter of type (I). The notion of the radical of a filter is defined and several characterizations of the radical of a filter are given. We show that in non conormal prelinear residuated skew lattice with element 0, infinitesimal elements are equal to intersection of all the maximal filters.