Superidentities in the variety of lattices
type: Item
instance of: scholarly article
title: Superidentities in the variety of lattices
zbMATH Open document ID: 0919.06003
DOI: 10.1007/BF02307222
published in: Mathematical Notes
publication date: 1997-07-22T00:00:00Z
review text: 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).
Mathematics Subject Classification ID: 06B20
Mathematics Subject Classification ID: 08A40
zbMATH DE Number: 1037786
zbMATH Keywords: varieties of lattices
zbMATH Keywords: hyperidentity
zbMATH Keywords: superidentity
zbMATH Keywords: polynomial superidentity
author: Yuri Movsisyan
reviewed by: Günther Eigenthaler
MaRDI profile type: Publication
cites work: Q3235339
description: scientific article; zbMATH DE number 1037786