## A note on complex algebras of semigroups (2003)

Venue: | Relational and Kleene-Algebraic Methods in Computer Science: Proc. 7th Int. Sem. Relational Methods in Computer Science and 2nd Int. Workshop Applications of Kleene Algebra |

Citations: | 2 - 1 self |

### BibTeX

@INPROCEEDINGS{Jipsen03anote,

author = {Peter Jipsen},

title = {A note on complex algebras of semigroups},

booktitle = {Relational and Kleene-Algebraic Methods in Computer Science: Proc. 7th Int. Sem. Relational Methods in Computer Science and 2nd Int. Workshop Applications of Kleene Algebra},

year = {2003},

pages = {171--177},

publisher = {Springer-Verlag}

}

### OpenURL

### Abstract

Abstract. The main result is that the variety generated by complex algebras of (commutative) semigroups is not finitely based. It is shown that this variety coincides with the variety generated by complex algebras of partial (commutative) semigroups. An example is given of an 8-element commutative Boolean semigroup that is not in this variety, and an analysis of all smaller Boolean semigroups shows that there is no smaller example. However, without associativity the situation is quite different: the variety generated by complex algebras of (commutative) binars is finitely based and is equal to the variety of all Boolean algebras with a (commutative) binary operator. A binar is a set A with a (total) binary operation ·, and in a partial binar this operation is allowed to be partial. We write x · y ∈ A to indicate that the product of x and y exists. A partial semigroup is an associative partial binar, i.e. for all x, y, z ∈ A, if (x · y) · z ∈ A or x · (y · z) ∈ A, then both terms exist and evaluate to the same element of A. Similarly, a commutative partial binar is a binar such that if x · y ∈ A then x · y = y · x ∈ A. Let (P)(C)Bn and (P)(C)Sg denote the class of all (partial) (commutative) groupoids and all (partial) (commutative) semigroups respectively. For A ∈ PBn the complex algebra of A is defined as Cm(A) = 〈P (A), ∪, ∅, ∩, A, \, ·〉, where X · Y = {x · y | x ∈ X, y ∈ Y and x · y exists} is the complex product of X, Y ∈ Cm(A). Algebras of the form Cm(A) are examples of Boolean algebras with a binary operator, i.e., algebras 〈B, ∨, 0, ∧, 1, ¬, ·〉 such that 〈B, ∨, 0, ∧, 1, ¬ 〉 is a Boolean algebra and · is a binary operation that distributes over finite (including empty) joins in each argument. A Boolean semigroup is a Boolean algebra with an associative binary operator. For a class K of algebras, Cm(K) denotes the class of all complex algebras of K, H(K) is the class of all homomorphic images of K, and V(K) is the variety generated by K, i.e., the smallest equationally defined class that contains K. The aim of this note is to contrast the equational theory of Cm((C)Bn) with that of Cm((C)Sg). It turns out that the former is finitely based while the latter is not.

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(Show Context)
Citation Context ...riety, and V(Cm((C)Sg)) = (C)Rel. Proof: The class (C)Rel is easily seen to be closed under subalgebras and products. The proof that (C)Rel is closed under homomorphic images is similar to a proof in =-=[2]-=- Theorem 5.5.10 that shows cylindric-relativized set algebras are a variety (see also [6] Theorem 1.5). Moreover, it is easy to see that (C)Rel ⊆ V(Cm(P(C)Sg)) since the algebra of all subsets of a tr... |

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(Show Context)
Citation Context ...}, and for each a ∈ S define Ra,0 = {〈a, a ′ 〉}, where a ′ = 〈a, 0〉. � Now the main result follows easily from the “representation theorem” that we have just established. Previously it was known from =-=[8]-=- that the variety generated by complex algebras of groups (i.e., the variety of group relation algebras) is not finitely based. In this case the analogous representation theorem states that every grou... |

34 |
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(Show Context)
Citation Context ...ve disjoint relations, so we take a step-by-step approach and use transfinite induction to build the Ra. A detailed discussion of this method for representing relation algebras can be found in [3] or =-=[4]-=-. Since our setting is somewhat different, and to avoid lengthy definitions, we take a rather informal approach here. To simplify the argument, we will arrange that all the relations are irreflexive a... |

29 | Step by step — building representations in algebraic logic
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(Show Context)
Citation Context ... not give disjoint relations, so we take a step-by-step approach and use transfinite induction to build the Ra. A detailed discussion of this method for representing relation algebras can be found in =-=[3]-=- or [4]. Since our setting is somewhat different, and to avoid lengthy definitions, we take a rather informal approach here. To simplify the argument, we will arrange that all the relations are irrefl... |

19 |
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Citation Context ...Rel (called R(∪, ∩, |, −) in [1]) is not finitely axiomatizable, and by the preceding result Rel = V(Cm(Sg)). Andreka’s result is proved using a sequence of finite commutative relation algebras (from =-=[7]-=-) such that the Boolean semigroup reducts of these algebras are not in Rel, but the ultraproduct is in CRel. It follows that CRel = V(Cm(CSg)) is also not finitely axiomatizable. � In fact one can fin... |

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Citation Context ...er subalgebras and products. The proof that (C)Rel is closed under homomorphic images is similar to a proof in [2] Theorem 5.5.10 that shows cylindric-relativized set algebras are a variety (see also =-=[6]-=- Theorem 1.5). Moreover, it is easy to see that (C)Rel ⊆ V(Cm(P(C)Sg)) since the algebra of all subsets of a transitive relation is the complex algebra of a partial semigroup, with ordered pairs as el... |

1 |
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(Show Context)
Citation Context ...orem states that every group relation algebra is representable, a result that follows directly from Cayley’s theorem for groups. Corollary 3 V(Cm(Sg)) and V(Cm(CSg)) are not finitely based. Proof: In =-=[1]-=- (Theorem 4) Andreka shows that the class Rel (called R(∪, ∩, |, −) in [1]) is not finitely axiomatizable, and by the preceding result Rel = V(Cm(Sg)). Andreka’s result is proved using a sequence of f... |

1 |
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(Show Context)
Citation Context ...e algebras A6, . . . , A11 can be embedded in complex algebras of finite semigroups. Finally we contrast the equational theory of complex algebras of semigroups with the following result adapted from =-=[5]-=- (Theorem 3.20). Theorem 5 Every Boolean algebra with a binary operator can be embedded in a member of Cm(PBn). If the operator is commutative, then the algebra can be embedded in a member of Cm(PCBn)... |

1 |
Complex algebras of semigroups, dissertation
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(Show Context)
Citation Context ... t5 = 0. The following result shows that there is no smaller example. Theorem 4 All four-element Boolean semigroups are in V(Cm(Sg)). Proof: P. Reich enumerated all four-element Boolean semigroups in =-=[9]-=-. There are a total of 50 (including isomorphic copies), which reduces to 28 if isomorphic copies are excluded. Of these, 6 are non-commutative with a corresponding “opposite” algebra, so only 22 need... |