Superidentities in the variety of lattices
A superidentity is a formula of the second-order language of the form: \(\forall X_1,\) \(\dots, X_m \forall x_1,\dots, x_n\) \((w_1= w_2)\), where \(X_1,\dots, X_m\) are function variables and \(x_1,\dots, x_n\) object variables in the terms \(w_1\), \(w_2\). A superidentity is said to hold in an algebra \(\langle Q;F\rangle\) (with \(Q\) as base set and \(F\) as set of operations) if it holds in case each object variable in it is replaced by an arbitrary element of \(Q\) and each function variable is replaced by an arbitrary operation of \(F\) (with corresponding arity). For a variety \(V\), a superidentity is said to be a superidentity of \(V\) if it holds in any algebra of \(V\). Consider the following superidentities: \[ (1)\quad X(x,x)= x;\quad (2)\quad X(x,y)= X(y,x);\quad (3)\quad X(x,X(y,z))= X(X(x,y), z); \] \[ (4)\qquad X(Y(X(x,y), z),Y(y,z))= Y(X(x,y),z); \] \[ (5)\qquad X(Y(x,X(y,z)), Y(y,z))= Y(X(x, Y(y,z)), X(y,z)); \] \[ (6)\qquad X(x,Y(y,z))= Y(X(x,y), X(x,z)). \] In the paper it is stated -- without proof -- that, for the variety \(L\) of lattices, any superidentity of \(L\) is a consequence of (1)--(4); for the variety \(M\) of modular lattices; any superidentity of \(M\) is a consequence of (1)--(5); and for the variety \(D\) of distributive lattices any superidentity of \(D\) is a consequence of (1)--(3), (6). Furthermore, the concept of polynomial superidentity is defined and similar results on polynomial superidentities are stated (for the varieties of distributive lattices and Boolean algebras).
scientific article; zbMATH DE number 1037786