We show that to every centred ternary relation T on a set A there can be assigned (in a non-unique way) a ternary operation t on A such that the identities satisfied by .AI t/ reflect relational properties of T . We classify ternary operations assigned to centred ternary relations and we show how the concepts of relational subsystems and homomorphisms are connected with subalgebras and homomorphisms of the assigned algebra .AI t/. We show that for ternary relations having a non-void median can be derived so-called median-like algebras .AI t/ which become median algebras if the median M T .a; b; c/ is a singleton for all a; b; c 2 A. Finally, we introduce certain algebras assigned to cyclically ordered sets.

The one-to-one correspondence between Boolean algebras and Boolean rings can be generalized to a mapping from partial ring-like algebras to bounded lattices with an antitone involution ( * -lattices), cf.[1], [2]. These partial algebras are known as partial generalized Boolean quasirings (pGBQRs) and can be used to construct various kinds of quantum logics (cf.[1] -[6]). To a large extent this is due to the fact that the partial operation ⊕ of pGBQRs can be extended almost arbitrarily to a full operation +. However, in view of Boolean rings and Boolean algebras it is evident to demand that + should be a term built up by the operations of the corresponding * -lattice. How this can be achieved is investigated as well for the case that the underlying lattice is an arbitrary * -lattice as for the special case of de Morgan algebras. Moreover, we study various classes of pGBQRs such that + is associative.Mathematics Subject Classification: 08A40, 06C15, 06D30

We generalize the one-to-one correspondence between Boolean algebras and Boolean rings to so-called difference lattices and commutative strong difference ring-like algebras. Moreover, we show that difference ring-like algebras induce some sort of symmetric difference in corresponding posets.

We present an easy construction producing a Kleene lattice K = (K, ⊔, ⊓, ′ ) from an arbitrary distributive lattice L and a non-empty subset of L. We show that L can be embedded into K and compute |K| under certain additional assumptions. We prove that every finite chain considered as a Kleene lattice can be represented in this way and that this construction preserves direct products. Moreover, we demonstrate that certain Kleene lattices that are ordinal sums of distributive lattices are representable. Finally, we prove that not every Kleene lattice is representable.

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