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Distributive lattices are well known to be precisely those lattices in which the relative complements are unique. This is the famous statement that a lattice is distributive if and only if it does not contain the two 5-element lattices M5 ands N5 as sublattices. There are no such non-trivial theorems in groups or rings. For example, there is no finite group H such that a group G is commutative if and only if G does not contain H as a subgroup. This phenomenon of characterizing a variety of algebras using the concept of "forbidden" subalgebras is kind of unique to lattice theory. However, generalizations of lattice varieties may also enjoy such "forbidden- algebra" theorems. Several natural generalziations of lattice varieties are known. Remove the associative laws and we get the equational class of weakly-associative lattices pioneered by Ervin Fried, George Gratzer and Bob Quackenbush. Remove the commutative laws to get the variety of non-commutative lattices pioneered by Jonathan Leech and his collaborators. Finally, retaining both associativity and commutativity but eliminating the two absorption laws, we arrive at the concept of quasilattices studied by R. Padmanabhan, J. Plonka and P. Penner. In this paper we study the property of uniqueness of relative complementation in the variety of all quasilattices. Exactly as in the case of lattices, a quasilattice satisfies the property URC: x v z = y v z and x ^ z = y ^ z ==> y = z if and only if it is a distributive lattice. We give two proofs: a second order proof using the structure of subdirectly irreducible quasilattices as well as a strictly first-order proof employing an automated theorem-prover. We prove a similar theorem for quasilattices not containing N5. Jonathan Leech, Michael Kinyon and their collaborators have studied a similar problem for the variety of non-commutative lattices. Of course, M5 and N5 are just two examples of forbidden lattices. Every finite subdirectly irreducible sublattice of a free lattice qualifies to be a candidate for being forbidden and this powerful concept goes back to Ralph MacKenzie in his seminal paper of 1972. Hence there are infinitely many such theorems to be proved (or to be disproved) in this direction. Some concrete open problems will be mentioned.

References:

1. Lakser, H.; Padmanabhan, R.; Platt, C. R., Subdirect decomposition of Plonka sums. Duke Math. J. 39 (1972), 485--488.
2. Padmanabhan, R.; Penner, P. Structures of free n-quasilattices. Algebra Colloq. 6 (1999), no. 3, 249--260.
3. Padmanabhan, R. Regular identities in lattices. Trans. Amer. Math. Soc. 158 1971 179--188. (Reviewer: N. C. A. da Costa) 06.30
4. Fried, E.; Gratzer, G.; Quackenbush, R. The equational class generated by weakly associative lattices with the unique bound property. Ann. Univ. Sci. Budapest. Eotvos Sect. Math. 22/23 (1979/80), 205--211.