Natural duality

Definition

Let $M$ be a finite algebra and let $\mathcal{A} = \mathbf{ISP}(M)$ be the quasivariety it generates. An alter ego of $M$ is a structure $\underset{\sim}{M} = (M; G, H, R, \mathcal{T})$ on the same set, where $\mathcal{T}$ is the discrete topology and:

$G$ is a set of operations $M^n \to M$ that are homomorphisms from $M^n$ to $M$ ;
$H$ is a set of partial operations whose domains are subalgebras of powers $M^n$ and which are homomorphisms on them;
$R$ is a set of relations $r \subseteq M^n$ that are subalgebras of $M^n$ .

For $A \in \mathcal{A}$ , the set $D(A)$ of homomorphisms $A \to M$ is a closed substructure of the power $\underset{\sim}{M}^A$ with the product topology. Let $X$ be a closed substructure of a nonzero power of $\underset{\sim}{M}$ . The set $E(X)$ of continuous maps $X \to M$ that preserve $G$ , $H$ and $R$ is a subalgebra of $M^X$ . Each $A$ has the evaluation homomorphism $e_A \colon A \to E(D(A))$ , $e_A(a)(x) = x(a)$ . The alter ego $\underset{\sim}{M}$ yields a natural duality on $\mathcal{A}$ if $e_A$ is an isomorphism for every $A \in \mathcal{A}$ .

Remarks

The duality is full if, in addition, every closed substructure $X$ of a nonzero power of $\underset{\sim}{M}$ is isomorphic to $D(E(X))$ by the evaluation map. Then $D$ and $E$ form a dual equivalence of categories.

Example: let $M$ be the two-element Boolean algebra and let $\underset{\sim}{M}$ be the two-element set with the discrete topology and $G = H = R = \emptyset$ . It yields a natural duality on Boolean algebras: this is Stone's representation of a Boolean algebra as the clopen subsets of its Stone space. For the two-element bounded lattice, the alter ego with the order relation $\le$ on $\set{0, 1}$ yields Priestley duality.

Non-example: for the two-element bounded lattice $M$ , the alter ego with no structure besides the topology does not yield a duality. Take for $A$ the three-element chain $0 < a < 1$ . It has two homomorphisms to $M$ , sending $a$ to $0$ or to $1$ , so $D(A)$ is a two-point discrete space. That space has four continuous maps to $M$ , while $A$ has three elements, so $e_A$ is not onto. The relation $\le$ is a subalgebra of $M^2$ , and adding it cuts these maps down to the three order-preserving ones.

Tag	`2WADQVGPD4`
URI	`m2://2WADQVGPD4`
Kind	definition
Writing	W0 stub
MSC	08C20, 06D50
Defines	Natural duality
Uses	Quasivariety, Homomorphism, Direct product of algebras, Topological space

Natural duality

Definition

Remarks

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