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Superidentities in the variety of lattices. (English. Russian original) Zbl 0919.06003
Math. Notes 59, No. 6, 686-688 (1996); translation from Mat. Zametki 59, No. 6, 944-946 (1996).
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).

##### MSC:
 06B20 Varieties of lattices 08A40 Operations and polynomials in algebraic structures, primal algebras
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##### References:
  A. Church,Introduction to Mathematical Logic, Vol. 1, Princeton University Press, Princeton (1956). · Zbl 0073.24301  A. I. Maltsev,Algebraic Systems [in Russian], Nauka, Moscow (1970).  Yu. M. Movsisyan,An Introduction to the Theory of Algebras with Superidentities [in Russian], Erevan State University, Erevan (1986).  Yu. M. Movsisyan,Superidentities and Supervarieties in Algebras [in Russian], Erevan State University, Erevan (1990).
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