and Beyond

except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now know or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if the are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Contents Preface vii 1 Unary algebras and dualisability 1 1.1 Quasi-varieties generated by finite algebras 1 1.2 Two examples 3 1.3 An introduction to dualisability 14 1.4 A dualisability toolkit 19 1.5 Full and strong dualisability 22 2 Binary homomorphisms and natural dualities 25 2.1 Lattice operations 26 2.2 Operations with strong idempotents 29 2.3 Dualisable term retracts 40 2.4 Group operations 46 3 The complexity of dualisability: three-element unary algebras 51 3.1 Dividing unary algebras into petals 54 3.2 One-kernel unary algebras 60 3.3 Dualisable two-kernel three-element unary algebras 69 3.4 Non-dualisable three-element unary algebras 80 3.5 Finite unars are dualisable 86 vi Contents 4 Full and strong dualisability: three-element unary algebras 89 4.1 Zero-kernel and one-kernel unary algebras revisited 92 4.2 Nice two-kernel three-element unary algebras 97 4.3 Three-element unary algebras that are not strongly dualisable 108 4.4 Three-element unary algebras that are not fully dualisable 119 5 Dualisability and algebraic constructions 135 5.1 Coproducts of unary algebras 138 5.2 Term retractions and p-semilattices 149 5.3 Building non-dualisable algebras from duaUsable ones 155 6 Dualisability and clones 163 6.1 Distant unions 165 6.2 Alternating chains of clones 169 6.3 Adding constants 174 6.4 A non-dualisable entropic algebra 177 7 Inherent dualisability 179 7.1 Embeddings into non-dualisable algebras 181 7.2 Linear unary algebras are dualisable 187 7.3 Linear unary algebras are strongly dualisable 201 Epilogue 207 Appendix: Strong dualisability 211 A.l Term-closed sets 212 A.2 Term closure via duals 214 A.3 The Inverse Limit Lemma at work 218 A.4 The height of a finite algebra 223 A.5 Reducing height to the finite level 231 A.6 From height to rank 235 A.7 Some examples 241 References 251 Notation 255 Index 259 Preface In mathematics, there are many methods for translating a difficult problem into an easier problem in a completely different setting. For example: • a logarithm function translates the difficult problem of multiplying two positive numbers into …