## Universal Algebra and CoalgebraThe purpose of this book is to study the structures needed to model objects in universal algebra, universal coalgebra and theoretical computer science. Universal algebra is used to describe different kinds of algebraic structures, while coalgebras are used to model state-based machines in computer science.The connection between algebras and coalgebras provides a way to connect static data-oriented systems with dynamical behavior-oriented systems. Algebras are used to describe data types and coalgebras describe abstract systems or machines.The book presents a clear overview of the area, from which further study may proceed. |

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### Contents

Introduction | 1 |

1 Algebras and Identities | 7 |

2 Statebased Systems | 35 |

3 Basic Concepts from Category Theory | 69 |

4 FCoalgebras | 89 |

5 FAlgebras | 113 |

6 F1 F2Coalgebras | 129 |

7 Terminal Coalgebras | 145 |

9 Coalgebras of Type | 167 |

10 Clones of Operations and Cooperations | 187 |

11 Semigroups of Operations and Cooperations | 213 |

12 Cohyperidentities and Msolid Classes of Coalgebras | 251 |

265 | |

Glossary | 271 |

273 | |

8 Cofree Fcoalgebras and Coequations | 155 |

### Common terms and phrases

Alg(r arbitrary automaton base set behavioural equivalence bijective binary operation binary relation bisimulation black box Boolean co-operations Boolean operations called category Set Chapter classes of algebras clones of Boolean co-operation symbol coalgebra homomorphism coalgebras of type coequalizer coidentity commutes cOn(A congruence relation consider constant semigroup Corollary coterms of type covarieties defined definition denote diagram in Figure disjoint epimorphism equation equivalence relation exists F-algebras F-coalgebras ff)iei finite set free algebra functor F Galois connection homomorphic images idempotent identity injective input isomorphic Kerf Kerg kernel Kripke structures lattice Lemma Let F morphism n-ary n-ary Boolean n-ary co-operations n-ary operation natural transformation non-empty notation objects output pair Proof properties Proposition Prove pullbacks pushout right-zero semigroup satisfied Section semigroup SetF state-based systems structural mapping subalgebra subsemigroup subset surjective terminal coalgebra Theorem tree automata unary uniquely determined universal algebra WT(X