A binary equality language is the equality language of two binary homomorphisms, that is, the set

Eq(g,h)={w : g(w)=h(w)},

where *g,h: {a,b}; ^{*} → Δ^{*}*.

The binary equality languages constitute the simplest non-trivial class of equality languages.

We say that the homomorphism *g: {a,b}; ^{*} → Δ^{*}* is periodic if

In this case *Eq(g,h)={ε}* or *Eq(g,h)={ε}∪{α;|α| _{a}∕|α|_{b}=k}* for some

In this case, up to the exchange of letters *a* and *b*, the equality language is the following: *Eq(g,h)={a ^{i}ba^{j}}^{*}*, for some

In this case *Eq(g,h)={α , β} ^{*}* for some, possibly empty, words

If both words *α* and *β* are non-empty, then, up to the exchange of the letters *a* and *b* we obtain *Eq(g,h)={a ^{i}b , ba^{i}}^{*}*.

If *Eq(g,h)* is generated by a single word, the exact structure of the equality language is still unknown. For more details see Binary equality sets with a single generator.

-- JanaHadravova - 18 Jun 2012

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Topic revision: r2 - 2012-06-22 - JeffreyShallit

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