Quasi-primal algebra

Builds on Ternary discriminator Term operation Finite set

Definition

Let $A$ be an algebra whose universe is a nonempty finite set. The algebra $A$ is quasi-primal if the ternary discriminator $t$ on $A$ is a term operation of $A$ . Here $t(x, y, z) = z$ if $x = y$ , and $t(x, y, z) = x$ if $x \ne y$ .

Remarks

Example: every finite field $F$ with $q$ elements, as a ring, is quasi-primal. Check: the term $x + (z - x)\bigl(1 - (x - y)^{q - 1}\bigr)$ gives $z$ when $x = y$ , and gives $x$ otherwise since then $(x - y)^{q - 1} = 1$ . Non-example: the two-element lattice $(\set{0, 1}, \wedge, \vee)$ is not quasi-primal. Its term operations are monotone, but the discriminator is not, since $t(0, 0, 1) = 1$ while $t(0, 1, 1) = 0$ . A finite algebra is quasi-primal if and only if it generates an arithmetical variety and every nontrivial subalgebra of it is simple. This characterization is due to Alden Pixley.

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Tag	`DJEK11CXXJ`
URI	`m2://DJEK11CXXJ`
Kind	definition
Writing	W0 stub
MSC	08A40, 08B05
Defines	Quasi-primal algebra
Uses	Ternary discriminator, Term operation, Finite set

Quasi-primal algebra

Definition

Remarks

Reading path

Further reading

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