## Abstract

It has been known since the 1960s that a class of algebras is axiomatisable and closed under subalgebras and homomorphic images if and only if it is defined by a system of disjunctive identities, i.e. identities of the form u1=v1∨u2=v2∨⋯∨uk=vk. However, given a set of such identities, it seems often very hard to describe the corresponding class of algebras, even in the case of semigroups.

Various modifications of disjunctive identities have been introduced, such as inclusive identities and collective identities. Let FX be a free semigroup over the algebra X and let u∈FX, U={u1,⋯,uk}⊆FX and V={v1,⋯,vk}⊆FX. Then u∈V is an inclusive identity and U=V is a collective identity. E. S. Lyapin [in Semigroups and their homomorphisms (Russian), 39–52, Ross. Gos. Ped. Univ., Leningrad, 1991; MR1171939] showed that these two concepts coincide, since every collective identity is equivalent to a system of inclusive identitities, and vice versa. In the class of groups these are also equivalent to disjunctive identities, but it is shown in this paper that this is not true in the class of semigroups. Examples are also given of a finite group without a finite basis of collective identities and a completely 0-simple semigroup with a finite basis of identities but without a finite system of disjunctive identities (such a semigroup is necessarily infinite).

Various modifications of disjunctive identities have been introduced, such as inclusive identities and collective identities. Let FX be a free semigroup over the algebra X and let u∈FX, U={u1,⋯,uk}⊆FX and V={v1,⋯,vk}⊆FX. Then u∈V is an inclusive identity and U=V is a collective identity. E. S. Lyapin [in Semigroups and their homomorphisms (Russian), 39–52, Ross. Gos. Ped. Univ., Leningrad, 1991; MR1171939] showed that these two concepts coincide, since every collective identity is equivalent to a system of inclusive identitities, and vice versa. In the class of groups these are also equivalent to disjunctive identities, but it is shown in this paper that this is not true in the class of semigroups. Examples are also given of a finite group without a finite basis of collective identities and a completely 0-simple semigroup with a finite basis of identities but without a finite system of disjunctive identities (such a semigroup is necessarily infinite).

Original language | English |
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Pages (from-to) | 124-140 |

Number of pages | 17 |

Journal | Algebra Universalis |

Volume | 35 |

Issue number | 1 |

State | Published - 1996 |