Congruence-preserving extensions of lattices
by
E. Tamas Schmidt

Let L and K be lattices. If L is a sublattice of K, we call K an extension of L. If K is an extension of L, \Theta is a congruence of L, and \phi is a congruence of K, then \Phi is an extension of \Theta to K iff the restriction of \Phi to L equals \Theta. We call K a congruence-preserving extension of L iff every congruence of L has exactly one extension to K. Let \phi be an embedding of L into K. If K is a congruence-preserving extension of L\phi, then we call \phi a congruence-preserving embedding of L into K.

In 1992, M. Tischendorf [7] verified that every finite lattice has a congruence-preserving embedding to a finite atomistic lattice.

A lattice L is sectionally complemented if for every b >= a in L, there is an element c that is the complement of a in the interval [0, b]. In 1996, G Grätzer and the author [5] proved the following:

In [4], G. Grätzer and E. T. Schmidt raised the following question:

Is it true that every lattice L with more than one element has a proper congruence-preserving extension K?

G. Grätzer and F. Wehrung [5] introduced the lattice tensor product, A \boxtimes B, of the lattices A and B. It is proved that for any finite lattice A, we can ``coordinatize'' A \boxtimes B, that is, represent A \boxtimes B as a subset A<B> of B^A.

They proved the isomorphism Con (A<B >) =~ (Con (A))<Con(B)>, which is a special case of a result of G. Grätzer and F. Wehrung and a generalization of a earlier result of G. Grätzer, H. Lakser, and R. W. Quackenbush.

Theorem 1. Every lattice L with more than one element has a proper congruence-preserving extension K.

We call the lattice L regular, if whenever \Theta and \Phi are congruences of L and \Theta and \Phi share a congruence class, then \Theta = \Phi.

We have proved the following result:

Theorem 2. Every lattice L has a congruence-preserving embedding into a regular lattice [L\tilde].

Atomistic lattices need not to be regular, the seven element semimodular, nonmodular lattice verifies this. On the other hand every sectionally complemented lattice is regular. Let us remark that M. Ploscica, J. Tma, and F. Wehrung [6] pointed out that not every bounded lattice admits a congruence-preserving extension into a sectionally complemented lattice.

A lattice L is congruence-finite, if Con(L) is finite; it is \omega-congruence-finite, if L can be written as a union, of an increasing sequence of congruence-finite sublattices of L.

G. Grätzer, H. Lakser and F. Wehrung proved [7]:

Theorem 3. Every \omega-congruence-finite lattice K has a \omega-congruence-finite, relatively complemented congruence-preserving extension L. Furthermore, if K has a zero, then L can be taken to have the same zero.

The last result is due to Wehrung [9], where Comp(K) denotes the semilattice of all compact congruence relations:

Theorem 4. Every lattice K such that Comp(K) is a lattice admits a congruence-preserving extension into a relatively complemented lattice.

References

[1] G. Grätzer, H. Lakser, and F. Wehrung, Congruence amalgamation of lattices, manuscript, 1998.

[2] G. Grätzer and E. T. Schmidt, A lattice construction and congruence-preserving extensions, Acta Math. Hungar. 66 (1995), 275-288.

[3] G. Grätzer and E. T. Schmidt, Congruence-preserving extensions of finite lattices to sectionally complemented lattices, Proc. Amer. Math. Soc. 127 (1999), 1903-1915.

[4] G. Grätzer and E. T. Schmidt, Regular congruence-preserving extensions. To appear in Algebra Universalis.

[5] G. Grätzer and E. T. Schmidt, Proper congruence-preserving extensions of lattices, Acta Math. Hungar. 85 (1999), 175-185.

[6] M. Ploscica, J. T uma and F. Wehrung, Congruence lattices of free lattices in non-distributive varieties, Colloq. Math. 76, (1998), 269-278.

[7] M. Tischendorf, The representation problem for algebraic distributive lattices, Ph.D. thesis, Fachbereich Mathematik der Technischen Hochschule Darmstadt, Darmstadt, 1992.

[8] F. Wehrung, Join-semilattices with two-dimensional congruence amalgamations, manuscript, 2000.

Date received: April 15, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Mathematical Conference Abstracts. Document # cagl-09.