# Bounded Lattice Expansions

@article{Gehrke2001BoundedLE, title={Bounded Lattice Expansions}, author={Mai Gehrke and John Harding}, journal={Journal of Algebra}, year={2001}, volume={238}, pages={345-371} }

Abstract The notion of a canonical extension of a lattice with additional operations is introduced. Both a concrete description and an abstract characterization of this extension are given. It is shown that this extension is functorial when applied to lattices whose additional operations are either order preserving or reversing, in each coordinate, and various results involving the preservation of identities under canonical extensions are established.

## 212 Citations

### A View of Canonical Extension

- MathematicsTbiLLC
- 2009

These lattices are doubly algebraic lattices and their interval topologies agree with their double Scott topologies and make them Priestley topological algebras.

### A note on profinite completions and canonical extensions

- Mathematics
- 2010

J. Harding has proved that the profinite limit of an algebra A in a finitely generated variety of monotone lattice expansions coincides with its canonical extension. In this note we drop the…

### Connecting the profinite completion and the canonical extension using duality

- Mathematics
- 2006

We show using duality and category theory that the profinite completion Â of a bounded distributive lattice expansion A is a homomorphic image of the canonical extension Aσ . Moreover the natural…

### Canonical and natural extensions in nitely-generated varieties of lattice-based algebras

- Mathematics
- 2010

The paper investigates completions in the context of nitely-generated latticebased varieties of algebras. In particular the structure of canonical extensions in such varieties is explored, and the…

### Canonical extensions of ordered algebraic structures and relational completeness of some substructural logics∗

- Mathematics
- 2005

In this paper we introduce canonical extensions of partially ordered sets and monotone maps and a corresponding discrete duality. We then use these to give a uniform treatment of completeness of…

### MacNeille completions and canonical extensions

- Mathematics
- 2004

The main technical construction reveals that the canonical extension of a monotone bounded lattice expansion can be embedded in the MacNeille completion of any sufficiently saturated elementary extension of the original structure.

### On profinite completions and canonical extensions

- Mathematics
- 2006

Abstract.We show that if a variety V of monotone lattice expansions is finitely generated, then profinite completions agree with canonical extensions on V. The converse holds for varieties of finite…

### A Fresh Perspective on Canonical Extensions for Bounded Lattices

- MathematicsAppl. Categorical Struct.
- 2013

A construction of canonical extension valid for all bounded lattices, which is shown to be functorial, with the property that the canonical extension functor decomposes as the composite of two functors, each of which acts on morphisms by composition, in the manner of hom-functors.

### Canonical extensions and relational completeness of some substructural logics*

- MathematicsJournal of Symbolic Logic
- 2005

Abstract In this paper we introduce canonical extensions of partially ordered sets and monotone maps and a corresponding discrete duality. We then use these to give a uniform treatment of…

## References

SHOWING 1-10 OF 35 REFERENCES

### Stone duality for lattices

- Mathematics
- 1997

Abstract. We present a new topological representation and Stone-type duality for general lattices. The dual objects of lattices are triples
$ (X, \perp, Y) $, where X, Y are the filter and ideal…

### Dualities for varieties of distributive lattices with unary operations II

- Mathematics
- 1998

This paper extends to the general setting of [11], [25] procedures presented earlier for varieties of Ockham algebras. Given a suitable finitely generated variety A of distributive-lattice-ordered…

### Duality and Canonical Extensions of Bounded Distributive Lattices with Operators, and Applications to the Semantics of Non-Classical Logics II

- Mathematics, PhilosophyStud Logica
- 2000

This paper considers logics that are sound and complete with respect to varieties of distributive lattices with certain classes of well-behaved operators for which a Priestley-style duality holds, and presents a way of constructing topological and non-topological Kripke-style models for these types of logics.

### Duality for Lattice-Ordered Algebras and for Normal Algebraizable Logics

- MathematicsStud Logica
- 1997

A new topological representation for general lattices is presented, identifying them as the lattices of stable compact-opens of their dual Stone spaces (stability refering to a closure operator on subsets) and extended to a full duality.

### The lattice of modal logics: an algebraic investigation

- Mathematics, PhilosophyJournal of Symbolic Logic
- 1980

It is shown that the degree of incompleteness with respect to Kripke semantics of any modal logic containing the axiom □ p → P or containing an axiom of the form □ m p ↔□ m +1 p for some natural number m is .

### Varieties of distributive lattices with unary operations I

- MathematicsJournal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics
- 1997

Abstract A unified study is undertaken of finitely generated varieties HSP () of distributive lattices with unary operations, extending work of Cornish. The generating algebra () is assusmed to be of…

### A Survey of Boolean Algebras with Operators

- Mathematics
- 1993

The purpose of this survey is to call attention to the unifying role of the concept of a Boolean algebra with operators. The first chapter contains a brief history of this concept and a list of…

### Some kinds of modal completeness

- Philosophy
- 1980

In the modal literature various notions of “completeness” have been studied for normal modal logics. Four of these are defined here, viz. (plain) completeness, first-order completeness, canonicity…