Binary hyperidentities of lattices

Abstract

An equational identity of a given type τ involves two kinds of symbols: individual variables and the operation symbols. For example, the distributive identityδ: x ⋅ (y + z) = x ⋅ y + x ⋅ z has three variable symbols {x, y, z} and two operation symbols {+, ⋅}. Here the variables range over all the elements of the base set while the two operation symbols are fixed. However, we shall say that an identity ishypersatisfied by a varietyV if, whenever we also allow the operation symbols to range over all polynomials of appropriate arity, the resulting identities are all satisfied byV in the usual sense. For example, the ring of integers 〈Z; +, ⋅〉 satisfies the above distributive law, but it does not hypersatisfy the same formal law because, e.g., the identityx + (y ⋅ z) = (x + y) ⋅ (x + z) is not valid. By contrast, δis hypersatisfied by the variety of all distributive lattices and is thus referred to as a distributive latticehyperidentity. Thus a hyperidentity may be viewed as an equational scheme for writing a class of identities of a given type and the original identities themselves are obtained as special cases by substituting specific polynomials of appropriate arity for the operation symbols in the scheme. In this paper, we provide afinite equational scheme which is a basis for the set of all binary lattice hyperidentities of type 〈2, 2, ⋯〉.