If the lattice is defined in terms of the order instead, i.e. (A, ≤) is a bounded partial order with a least upper bound and greatest lower bound for every pair of elements, and the meet and join operations so defined satisfy the distributive law, then the complementation can also be defined as an involutive anti-automorphism, that is, a structure A = (A, ≤, ¬) such that: De Morgan algebras were introduced by Grigore Moisil[1][2] around 1935,[2] although without the restriction of having a 0 and a 1.
[3] They were then variously called quasi-boolean algebras in the Polish school, e.g. by Rasiowa and also distributive i-lattices by J.
They have been further studied in the Argentinian algebraic logic school of Antonio Monteiro.
[1][2] De Morgan algebras are important for the study of the mathematical aspects of fuzzy logic.
If the pseudocomplement satisfies the law of the excluded middle, the resulting algebra is also Boolean.