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Stream: theory: mathematics

Topic: Bands: xx=x

Jan Pax (Jul 03 2026 at 18:26):

Why in semigroups satisfying $xx=x$ it holds that
$xyzyx=xyxzxyx$ Use no advanced theorem, just rewriting rules like $AA\to A$.

Amar Hadzihasanovic (Jul 03 2026 at 19:16):

It does not hold in general bands.
It does hold in regular bands, which satisfy $xyxzx = xyzx$, by the equational chain
$(xyxzx)yx = [x(yzx]y)x = x[yz(yxy]x) = xyz[yx]$
where round brackets denote the part next rewritten and square brackets the one that's been rewritten; the first two equations by regularity, the third by idempotence.

Jan Pax (Jul 03 2026 at 19:25):

doi.org/10.1007/978-94-009-3839-7_6 Great. But, why in that link it holds that if c(w)={x} then s(w) is empty ?

Amar Hadzihasanovic (Jul 03 2026 at 19:41):

If I understand correctly, if $w$ is a string of elements of $X$ (called "variables"),

• $c(w)$ is "the set of all variables appearing in $w$",
• $\sigma(w)$ is the "variable that appears last in $w$",
• $s(w)$ is the “prefix of $w$ before the first appearance of $\sigma(w)$”

So if $c(w) = \{x\}$ it means that $x$ is the only variable appearing in $w$, that is, $w = x\cdots x$; so $x$ is (trivially) the “last variable”; and because it appears in the first position, there is no “prefix before the appearance of $x$”, so $s(w)$ is the empty string.

Amar Hadzihasanovic (Jul 03 2026 at 19:43):

For example if $w = xyxzx$ we have $c(w) = \{ x, y, z \}$, $\sigma(w) = z$, and $s(w) = xyx$.

Amar Hadzihasanovic (Jul 03 2026 at 20:00):

If you need an explicit counterexample to the equation you gave, there is a 5-element band where it does not hold: let its set of elements be $S = \{ 0, 1, 2, a, b \}$, and multiplcation defined by

• $xj = j$ for all $x \in S$ and $j \in \{ 0, 1, 2 \}$,
• $0a = 1a = 0b = 0$,
• $2a = 1b = 2b = 2$,
• $aa = ab = a$,
• $ba = bb = b$.

Then $ab1ba = 1ba = 2$ but $aba1aba = 1aba = 0$.