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Varieties of Interior Algebras [PhD Thesis]

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This is a doctoral dissertation of Wim Blok accomplished under supervision of prof. Dr. Ph.Dwinger in 1976. It is one of the outstanding dissertations on modal logic, although the title does not explicitly refer to modal logic. Abstract. We study (generalized) Boolean algebras endowed with an interior operator, called (generalized) interior algebras. Particular attention is paid to the structure of the free (generalized) interior algebra on a finite number of generators. Free objects in some varieties of (generalized) interior algebras are determined. Using methods of a universal algebraic nature we investigate the lattice of varieties of interior algebras.

Author(s): W.J.Blok
Publisher: Universiteit van Amsterdam
Year: 1976

Language: English
Commentary: DjVu'ed, OCR'ed, TOC by Envoy
Pages: 275
City: Amsterdam

Cover ......Page 1
Abstract ......Page 3
Acknowledgements ......Page 6
Table of contents ......Page 7
1. Some remarks on the subject and its history ......Page 10
2. Relation to modal logic ......Page 12
3. The subject matter of the paper ......Page 16
1. Universal algebra ......Page 22
2. Lattices ......Page 32
1. Generalized interior algebras: definitions and basic properties ......Page 37
2. Interior algebras: definition, basic properties and relation with generalized interior algebras ......Page 45
3. Two infinite interior algegras generated by one element ......Page 51
4. Principal ideals in finitely generated free algebras in Bi and Bi- ......Page 57
5. Subalgebras of finitely generated free algebras in Bi and Bi- ......Page 71
6. Functional freeness of finitely generated algebras in Bi and Bi- ......Page 78
7. Some remarks on free products, injectives and weakly projectives in Bi and Bi- ......Page 91
Chapter II. On Some Varieties of (Generalized) Interior Algebras ......Page 106
1. Relations between subvarieties of Bi and H, Bi and H-, Bi and Bi- ......Page 107
2. The variety generated by all (generalized) interior *-algebras ......Page 116
3. The free algebra on one generator in Bi-* ......Page 125
4. Injectives and projectives in Bi* and Bi-* ......Page 133
5. Varieties generated by (generalized) interior algebras whose lattices of open elements are chains ......Page 140
6. Finitely generated free objects in Mn- and Mn, for n in N ......Page 149
7. Free objects in M- and M ......Page 166
Chapter III. The Lattice of Subvarieties of Bi ......Page 173
1. General results ......Page 174
2. Equations defining subvarieties of Bi ......Page 178
3. Varieties associated with finite subdirectly irreducibles ......Page 188
4. Locally finite and finite varieties ......Page 199
5. The lattice of subvarieties of M ......Page 210
6. The lattice of subvarieties of (Bi:K3) ......Page 221
7. The relation between the lattices of subvarieties of Bi and H ......Page 230
8. On the cardinality of some sublattices of Omega ......Page 240
9. Subvarieties of Bi not generated by their finite members ......Page 250
References ......Page 259
Samenvatting ......Page 267
Subject index ......Page 269
Index of symbols ......Page 271
Stellingen bij het proefschrift "Varieties of interior algebras" ......Page 273