Journal of Formalized Mathematics
Volume 8, 1996
University of Bialystok
Copyright (c) 1996 Association of Mizar Users

## The Equational Characterization of Continuous Lattices

Mariusz Zynel
Warsaw University, Bialystok

### Summary.

The class of continuous lattices can be characterized by infinitary equations. Therefore, it is closed under the formation of subalgebras and homomorphic images. Following the terminology of [18] we introduce a continuous lattice subframe to be a sublattice closed under the formation of arbitrary infs and directed sups. This notion corresponds with a subalgebra of a continuous lattice in [16].\par The class of completely distributive lattices is also introduced in the paper. Such lattices are complete and satisfy the most restrictive type of the general distributivity law. Obviously each completely distributive lattice is a Heyting algebra. It was hard to find the best Mizar implementation of the complete distributivity equational condition (denoted by CD in [16]). The powerful and well developed Many Sorted Theory gives the most convenient way of this formalization. A set double indexed by $K$, introduced in the paper, corresponds with a family $\{x_{j,k}: j\in J, k\in K(j)\}$. It is defined to be a suitable many sorted function. Two special functors: $\rm Sups$ and $\rm Infs$ as counterparts of $\rm Sup$ and $\rm Inf$ respectively, introduced in [33], are also defined. Originally the equation in Definition~2.4 of [16, p. 58] looks as follows: $${\textstyle\bigwedge}_{j\in J} {\textstyle\bigvee}_{k\in K(j)} x_{j,k} = {\textstyle\bigvee}_{f\in M} {\textstyle\bigwedge}_{j\in J} x_{j,f(j)},$$ where $M$ is the set of functions defined on $J$ with values $f(j)\in K(j)$.

This work was partially supported by the Office of Naval Research Grant N00014-95-1-1336.

#### MML Identifier: WAYBEL_5

The terminology and notation used in this paper have been introduced in the following articles [25] [12] [31] [32] [15] [29] [9] [11] [10] [13] [1] [3] [24] [30] [4] [2] [8] [26] [20] [21] [27] [19] [23] [22] [5] [17] [33] [14] [6] [7] [28]

#### Contents (PDF format)

1. The Continuity of Lattices
2. Completely-Distributive Lattices
3. Sub--Frames of Continuous Lattices

#### Acknowledgments

The author is greatly indebted to Grzegorz Bancerek for his inspiring influence during the preparation of this paper. Thanks are also due to the whole Mizar team in Bia{\l}ystok for collaboration.

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Received October 25, 1996