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  "body": "\n## Definition\n\nLet $M$ be a finite algebra and let $\\mathcal{A} = \\mathbf{ISP}(M)$ be the [[quasivariety|WRXEKT4NQS]] it generates.\nAn *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|0CQHFPGJ45]] and:\n\n1. $G$ is a set of operations $M^n \\to M$ that are [[homomorphisms|KBXSZC6BV1]] from $M^n$ to $M$;\n2. $H$ is a set of partial operations whose domains are [[subalgebras|7CHG6SD365]] of powers $M^n$ and which are homomorphisms on them;\n3. $R$ is a set of relations $r \\subseteq M^n$ that are subalgebras of $M^n$.\n\nFor $A \\in \\mathcal{A}$, the set $D(A)$ of homomorphisms $A \\to M$ is a closed substructure of the power $\\underset{\\sim}{M}^A$ with the product topology.\nLet $X$ be a closed substructure of a nonzero power of $\\underset{\\sim}{M}$.\nThe set $E(X)$ of continuous maps $X \\to M$ that preserve $G$, $H$ and $R$ is a subalgebra of $M^X$.\nEach $A$ has the evaluation homomorphism $e_A \\colon A \\to E(D(A))$, $e_A(a)(x) = x(a)$.\nThe alter ego $\\underset{\\sim}{M}$ *yields a natural duality* on $\\mathcal{A}$ if $e_A$ is an isomorphism for every $A \\in \\mathcal{A}$.\n\n## Remarks\n\nThe 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.\nThen $D$ and $E$ form a dual equivalence of categories.\n\nExample: let $M$ be the two-element Boolean algebra and let $\\underset{\\sim}{M}$ be the two-element set with the discrete topology and $G = H = R = \\emptyset$.\nIt yields a natural duality on Boolean algebras: this is Stone's representation of a Boolean algebra as the clopen subsets of its Stone space.\nFor the two-element bounded lattice, the alter ego with the order relation $\\le$ on $\\set{0, 1}$ yields Priestley duality.\n\nNon-example: for the two-element bounded lattice $M$, the alter ego with no structure besides the topology does not yield a duality.\nTake for $A$ the three-element chain $0 \u003c a \u003c 1$.\nIt has two homomorphisms to $M$, sending $a$ to $0$ or to $1$, so $D(A)$ is a two-point discrete space.\nThat space has four continuous maps to $M$, while $A$ has three elements, so $e_A$ is not onto.\nThe relation $\\le$ is a subalgebra of $M^2$, and adding it cuts these maps down to the three order-preserving ones.\n",
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