## Morita equivalence of almost-primal clones (1996)

Venue: | J. Pure Appl. Algebra |

Citations: | 3 - 1 self |

### BibTeX

@ARTICLE{Bergman96moritaequivalence,

author = {Clifford Bergman and Joel Berman},

title = {Morita equivalence of almost-primal clones},

journal = {J. Pure Appl. Algebra},

year = {1996},

volume = {108},

pages = {175--201}

}

### OpenURL

### Abstract

Abstract. Two algebraic structures A and B are called categorically equivalent if there is a functor from the variety generated by A to the variety generated by B, carrying A to B, that is an equivalence of the varieties when viewed as categories. We characterize those algebras categorically equivalent to A when A is an algebra whose set of term operations is as large as possible subject to constraints placed on it by the subalgebra or congruence lattice of A, or the automorphism group of A. Two categories C and D are said to be equivalent if there are functors F: C → D and G: D → C such that the composite functors F ◦ G and G ◦ F are naturally isomorphic to the identities on D and C respectively. It is natural to ask whether some property of an object, morphism or an entire category is preserved under every equivalence of categories. Moreover, given an object (or morphism, or category), one might wish to characterize the class of objects obtained by applying all equivalences to that starting object. Any variety of algebras (that is, a class of algebras closed under the formation of subalgebra, product, and homomorphic image) forms a category, in which the morphisms

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Citation Context ...ry aj = p � σt1(aj),...,σtn(aj) � , p preserves R. Finally, we apply assumption (1) to obtain a term t ∈ Clon(A) extending p. Then t(σt1(x),...,σtn(x)) = x for all x ∈ A showing σ is invertible. � In =-=[1]-=- Baker and Pixley show that a finite algebra satisfies condition (1) of Theorem 2.1 if and only if it possesses a (k + 1)-ary near-unanimity term. (The reader may consult that paper for the definition... |

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Citation Context ...algebra that is term equivalent to B. The notion of categorical equivalence has been applied to algebraic structures numerous times in the literature. We mention papers of Davey and Werner [5], Freyd =-=[11]-=-, Isbell [15] and Wraith [30] in this regard. Recently in [18], R. McKenzie provided a powerful tool that is particularly well-suited to studying the behavior of “algebraic” properties under equivalen... |

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Citation Context ...c and thus A ≡c A(σ) ≡c B(σ ′ ) ≡c B. � Example 3.3. Let A be a k-element subalgebra-primal algebra having exactly one proper subalgebra, and suppose this subalgebra has cardinality m. Itisknown (see =-=[27]-=-) that Clo(A) is a co-atom in the lattice of clones on A. Ifk=3andm=2, then A is term equivalent to the 3-element ̷Lukasiewicz algebra ̷L and if k =2and m=1,thenAis term equivalent to the 2-element Bo... |

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Citation Context ...ppings S ↦→ S↾σ(A) and S ↦→ Sn . (2) For all B ∈ J (Sub A) there exists b ∈ B ∩ σ(A) with 〈b〉 = B. (3) S ∈ N(A) if and only if S ∩ σ(A) ∈ N(A(σ)). (4) For every X ⊆ σ(A), Sg A (X)↾ σ(A) =Sg A(σ) (X). =-=(5)-=- Con A ∼ = Con � A(σ) � � ∼ [n] = Con A � by θ ↦→ θ↾σ(A) and θ ↦→ θ [n] . These mappings preserve the permutability of congruences. (6) Aut(A) ∼ = Aut � A(σ) � � ∼ [n] = Aut A � by γ ↦→ γ↾σ(A) and γ ↦... |

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Citation Context ...operation on A that preserves every congruence of A is a term operation of A. Unlike the case for subalgebra-primal algebras for which we have the concrete version of the Birkhoff-Frink Theorem, (see =-=[3]-=- or [20, pg. 183]), there exist 0,1sublattices L of Eqv A with A finite for which there is no algebra A with universe AandCon A = L. In fact, a long-standing open question asks whether every finite la... |

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Citation Context ...is term equivalent to B. The notion of categorical equivalence has been applied to algebraic structures numerous times in the literature. We mention papers of Davey and Werner [5], Freyd [11], Isbell =-=[15]-=- and Wraith [30] in this regard. Recently in [18], R. McKenzie provided a powerful tool that is particularly well-suited to studying the behavior of “algebraic” properties under equivalence and to giv... |

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Citation Context ...lgebra isomorphic to Cn = � {0, 1,...,n},¬,→,0,n � in which ¬x = n − x and x → y =max{y−x, 0}. These algebras are the natural generalization of the 3-element ̷Lukasiewicz algebras in Example 3.3. See =-=[9]-=- and [12]. Each Cn is term equivalent to a bounded, commutative BCK chain as studied by Traczyk [28]. From Denecke [6, pg. 134] we see that each Cn is subalgebraprimal. The lattice Sub(Cn) is isomorph... |

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Citation Context ...orresponding to ∅) has been added. The unique atom in Sub(Cn) has precisely two elements, {0,n}. Thus, by Theorem 3.2, we have Cn ≡c Cm if and only if n and m have isomorphic lattices of divisors. In =-=[21]-=-, Murskiĭ proved that almost all algebras of a sufficiently rich similarity type are subalgebra-primal. We conclude this section by considering his theorem in conjunction with Theorem 3.2. Let τ be a ... |

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Citation Context ... lattice is isomorphic to the congruence lattice of a finite algebra. On the other hand, if L is a finite, distributive 0,1-sublattice of Eqv A then Quackenbush and Wolk proved in [26] (see also [2], =-=[16]-=-) that L = Con A for an algebra A with universe A. In fact, if we form the algebra B = 〈A, P(L)〉 then B will be congruence primal and Con B = L. We focus on distributive 0,1-sublattices L of Eqv A, fo... |

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Citation Context ...e algebras are the natural generalization of the 3-element ̷Lukasiewicz algebras in Example 3.3. See [9] and [12]. Each Cn is term equivalent to a bounded, commutative BCK chain as studied by Traczyk =-=[28]-=-. From Denecke [6, pg. 134] we see that each Cn is subalgebraprimal. The lattice Sub(Cn) is isomorphic to the lattice of divisors of n in which a new bottom element (corresponding to ∅) has been added... |

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Citation Context ...der for their varieties of unitary modules to be equivalent as categories. Other instances of categorical equivalence of varieties have been discovered using the tools of duality theory as in [5] and =-=[17]-=-. There are also examples of objects characterized up to categorical equivalence in the literature. One of the most striking is a result of Hu’s [14]. A finite, nontrivial algebra A is called primal i... |

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Citation Context ...l equivalence has been applied to algebraic structures numerous times in the literature. We mention papers of Davey and Werner [5], Freyd [11], Isbell [15] and Wraith [30] in this regard. Recently in =-=[18]-=-, R. McKenzie provided a powerful tool that is particularly well-suited to studying the behavior of “algebraic” properties under equivalence and to giving a complete algebraic description of A/≡c for ... |

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Citation Context ...nt to B. The notion of categorical equivalence has been applied to algebraic structures numerous times in the literature. We mention papers of Davey and Werner [5], Freyd [11], Isbell [15] and Wraith =-=[30]-=- in this regard. Recently in [18], R. McKenzie provided a powerful tool that is particularly well-suited to studying the behavior of “algebraic” properties under equivalence and to giving a complete a... |

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Citation Context ...ed using the tools of duality theory as in [5] and [17]. There are also examples of objects characterized up to categorical equivalence in the literature. One of the most striking is a result of Hu’s =-=[14]-=-. A finite, nontrivial algebra A is called primal if every operation on the universe of A is a term operation of A. For example, the two-element Boolean algebra is primal. Hu’s theorem for primal alge... |

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Citation Context ...e literature, the finite subalgebraprimal algebras have been called semi-primal. Similarly, the finite members of the other two classes have been called hemi-primal and demi-primal, respectively. See =-=[22]-=-, [24] and [29]. The primary goal of this paper is to describe, up to categorical equivalence, all finite algebras that are subalgebra-primal, congruence-primal and arithmetical, or automorphism-prima... |

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Citation Context ...inite lattice is isomorphic to the congruence lattice of a finite algebra. On the other hand, if L is a finite, distributive 0,1-sublattice of Eqv A then Quackenbush and Wolk proved in [26] (see also =-=[2]-=-, [16]) that L = Con A for an algebra A with universe A. In fact, if we form the algebra B = 〈A, P(L)〉 then B will be congruence primal and Con B = L. We focus on distributive 0,1-sublattices L of Eqv... |

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Citation Context ...c to each of Sub � A(σ) � and Sub � A [n]� via the mappings S ↦→ S↾σ(A) and S ↦→ Sn . (2) For all B ∈ J (Sub A) there exists b ∈ B ∩ σ(A) with 〈b〉 = B. (3) S ∈ N(A) if and only if S ∩ σ(A) ∈ N(A(σ)). =-=(4)-=- For every X ⊆ σ(A), Sg A (X)↾ σ(A) =Sg A(σ) (X). (5) Con A ∼ = Con � A(σ) � � ∼ [n] = Con A � by θ ↦→ θ↾σ(A) and θ ↦→ θ [n] . These mappings preserve the permutability of congruences. (6) Aut(A) ∼ = ... |

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Citation Context ... ∈ N(A(σ)). (4) For every X ⊆ σ(A), Sg A (X)↾ σ(A) =Sg A(σ) (X). (5) Con A ∼ = Con � A(σ) � � ∼ [n] = Con A � by θ ↦→ θ↾σ(A) and θ ↦→ θ [n] . These mappings preserve the permutability of congruences. =-=(6)-=- Aut(A) ∼ = Aut � A(σ) � � ∼ [n] = Aut A � by γ ↦→ γ↾σ(A) and γ ↦→ γ [n] . Proof. First consider (4). Let X ⊆ σ(A) andletB=Sg A(σ) (X). Every element of B is of the form σt(x1,...,xm), with x1,...,xm ... |

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Citation Context ...nd if k =2and m=1,thenAis term equivalent to the 2-element Boolean ring without unit, R. From Theorems 3.1 and 3.2 we see that for every such A with m =1,A≡c Rand if m>1thenA≡c ̷L. Denecke and Lüders =-=[7]-=- have carried out a classification with respect to ≡c of the algebras corresponding to all the clones that are co-atoms in the lattice of clones on a finite set. 9sExample 3.4. For n ≥ 1ann-element Wa... |

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Citation Context ...τk| and Pr(P ; τ) = lim Pr(P ; τk) k→∞ if this limit exists. Thus, Pr(P ; τ) is the probability that an arbitrary finite algebra of type τ has property P . For a discussion of this concept see Freese =-=[10]-=-. We say “almost all” algebras of type τ have property P if Pr(P ; τ) =1. For an operation f on a set A, let Fix(f) ={i∈A:f(i,...,i)=i}.ForS⊆A we denote by C(A, S) the clone of all operations f on A h... |

1 |
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Citation Context ...somorphic to Cn = � {0, 1,...,n},¬,→,0,n � in which ¬x = n − x and x → y =max{y−x, 0}. These algebras are the natural generalization of the 3-element ̷Lukasiewicz algebras in Example 3.3. See [9] and =-=[12]-=-. Each Cn is term equivalent to a bounded, commutative BCK chain as studied by Traczyk [28]. From Denecke [6, pg. 134] we see that each Cn is subalgebraprimal. The lattice Sub(Cn) is isomorphic to the... |

1 |
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Citation Context ...type τ has at least one at least binary operation, then almost all finite algebras A of similarity type τ 10scontain C(A, A) inCloA. Thus, almost all algebras of type τ are subalgebraprimal. McKenzie =-=[19]-=-, in an exposition of Murskiĭ’s result, shows that if τ contains at least two operation symbols, at least one of which has arity ≥ 2, then almost all finite algebras of type τ are primal. Combining th... |

1 |
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Citation Context ...rature, the finite subalgebraprimal algebras have been called semi-primal. Similarly, the finite members of the other two classes have been called hemi-primal and demi-primal, respectively. See [22], =-=[24]-=- and [29]. The primary goal of this paper is to describe, up to categorical equivalence, all finite algebras that are subalgebra-primal, congruence-primal and arithmetical, or automorphism-primal. Obv... |

1 |
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Citation Context ...cular, no analogue of Theorem 4.2 is known.) If we require our automorphism-primal algebras to have no subalgebras we arrive at the notion of demi-primal algebra originally proposed by Quackenbush in =-=[25]-=-: we shall call a finite algebra A Q-demi-primal if A is automorphism-primal and no non-identity automorphism has a fixed point. When we restrict our attention to that class, we get an easily stated r... |

1 |
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(Show Context)
Citation Context ...whether every finite lattice is isomorphic to the congruence lattice of a finite algebra. On the other hand, if L is a finite, distributive 0,1-sublattice of Eqv A then Quackenbush and Wolk proved in =-=[26]-=- (see also [2], [16]) that L = Con A for an algebra A with universe A. In fact, if we form the algebra B = 〈A, P(L)〉 then B will be congruence primal and Con B = L. We focus on distributive 0,1-sublat... |