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AbstractNecessary and sufficient conditions will be found for quadruples of lattices to be isomorphic to lattices of weak, relative, strong subalgebras and initial segments, respectively, of one partial unary algebra. To this purpose we will start with a characterization of pairs of lattices that are weak and strong subalgebra lattices of one partial unary algebra, respectively. Next, we will describe the initial segment lattice of a partial unary algebra. Applying this result, pairs ofdoi:10.1515/dema-2013-0416 fatcat:on6v2h42abdvvbja54d5tjpw3u