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  "body": "\n## Definition\n\nLet $\\mathcal{K}$ be a class of algebras of the same signature.\nAssume that there is a ternary [[term|N9M2N27END]] $s(x, y, z)$ whose term operation on every $A \\in \\mathcal{K}$ is the [[ternary discriminator|72PG6XCQ7D]] on $A$.\nThe [[variety|AJNZJD8442]] $V(\\mathcal{K})$ generated by $\\mathcal{K}$ is then called a *discriminator variety*, with discriminator term $s$.\nA variety is a discriminator variety if it is of the form $V(\\mathcal{K})$ for such a class $\\mathcal{K}$.\n\n## Remarks\n\nExample: the variety of Boolean algebras is generated by the two-element Boolean algebra $\\mathbf{2}$.\nOn $\\mathbf{2}$, the term $s(x, y, z) = (x \\wedge (x \\oplus y)) \\vee (z \\wedge \\neg(x \\oplus y))$, with $x \\oplus y = (x \\wedge \\neg y) \\vee (\\neg x \\wedge y)$, is the discriminator.\nCheck: if $x = y$ then $x \\oplus y = 0$ and $s = z$; if $x \\neq y$ then $x \\oplus y = 1$ and $s = x$.\nNon-example: the variety of lattices is not a discriminator variety.\nA discriminator term $s$ satisfies $s(x, y, y) \\approx x$ and $s(x, x, y) \\approx y$ on $\\mathcal{K}$, hence on $V(\\mathcal{K})$.\nSo a discriminator variety is congruence-permutable, while the three-element chain is a lattice with non-permuting congruences.\nThe identities $s(x, y, y) \\approx x \\approx s(x, y, x)$ and $s(x, x, y) \\approx y$ show that $s$ is a Pixley term, so every discriminator variety is an [[arithmetical variety|GW78A05TNA]].\nThe term $s$ need not act as the discriminator on every member of $V(\\mathcal{K})$; for Boolean algebras it does so only on $\\mathbf{2}$ and the trivial algebra.\n",
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