Two identities for lattices, distributive lattices and modular lattices with a constant.
Saburo Tamura · Notre Dame Journal of Formal Logic · 1975
In his paper [3] J. A. Kalman has defined lattices using two identities and six variables.We shall define lattices using two identities and five variables in Theorem 1.In Theorem 2 we shall give an axiom system for lattices with 0 consisting of two identities.J. Sholander's axiom system for distributive lattices with 0 contains three identities (c/.,[5]), but our axiom system in Theorem 3 consists of two identities.In Theorem 4 we shall give a definition for distributive lattices with 1 in the Croisot-Sobociήski style (c/., [l] and [7]).Finally, as axiom system for modular lattices with 0 shall be given in Theorem 5.In the remarks, axiom systems for lattices, distributive lattices and modular lattices with two constants are given by three identities.