![](data:image/png;base64,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Estructura algebraica de los aut´omatas finitos y lenguajes 21
Proposici´on 4.4. Sea Aun subconjunto regular de Σ∗y sea k∈K, entonces kA es un subcon-
junto regular de Σ∗.
Demostraci´on. Sea Aun K-aut´omata tal que |A| =A, y sea k∈K. Consideremos el K-aut´omata
kA= (Q, Σ, E, kI, T ), donde (kI)q=kIq(Ivisto como un vector fila), entonces
|kA| =X
p,q∈Q
klpE∗
pqTq=kX
p,q∈Q
lpE∗
pqTq=k|A| =kA.
Proposici´on 4.5. Un subconjunto Ade Σ∗es regular si, y s´olo si, el subconjunto A0=A∩Σ+
tambi´en lo es.
Demostraci´on. Sea Aun K-aut´omata tal que |A| =A. Entonces, existe un K-aut´omata (normali-
zado) A0tal que |A0|=|A|∩Σ+=A∩Σ+=A0. Reciprocamente, supongamos que A∩Σ+=A0es
un subconjunto regular de Σ∗. Como A0(θ) = 0 (ya que Σ+(θ) = 0), podemos escribir A=kθ+A0
donde k=A(θ). Ahora, como θes un subconjunto regular de Σ∗, entonces por las proposiciones
4.1 y 4.4 se tiene que Aes regular.
Proposici´on 4.6. Si AyBson subconjuntos regulares de Σ∗, entonces AB tambi´en lo es.
Demostraci´on. Sea A=kθ +A0,B=lθ +B0, con A0=A∩Σ+,B0=B∩Σ+,k=A(θ) y
l=B(θ). Entonces, AB =klθ +kB0+lA0+A0B0. Basta probar que A0B0es regular. Para esto,
sean A= (Q1,Σ, E1, i1, t1) y B= (Q2,Σ, E2, i2, t2) dos K-aut´omatas normalizados que reconocen
aA0yB0respectivamente. Consideremos el K-aut´omata normalizado C= (Q, Σ, E, i1, t2), donde
Qes la uni´on disjunta de Q1yQ2, salvo la consideraci´on t1=i2. Luego un arco en Ces un arco
en Ao es un arco en B. As´ı, claramente
|C| =E∗
i1t2=E∗
1i1t1E∗
2i2t2=|A||B| =A0B0.
Teorema 4.2. Un subconjunto Ade Σ+es regular si, y s´olo si, existe un entero n > 1yEuna
matriz n×ncuyas entradas son subconjuntos de Σtal que A=E+
1n.
Demostraci´on. Supongamos que Aes un subconjunto regular de Σ+, y sea A= (Q, Σ, E, i, t)
un K-aut´omata normalizado que reconoce a A. Sin perdida de generalidad supongamos que
Q={1,· · · , n},con i= 1 y t=n. Como i6=tse tiene que n > 1. Ahora, por el corolario 4.1
se tiene que |A| =E∗
1n=E+
1n=A. Reciprocamente, si A=E+
1n, donde Ees una matriz n×n
de subconjuntos de Σ y n > 1, entonces poniendo Q={1,· · · , n},I= 1 y T=n, obtenemos un
K-aut´omata A= (Q, Σ, E, l, n) que como antes |A| =E+
1n=A.
Corolario 4.2. Si Ees una matriz n×nde subconjuntos de Σ, entonces para cualesquiera
1≤i, j ≤n, los subconjuntos E∗
ij yE+
ij son regulares.
Demostraci´on. Inmediato desde el teorema 4.2
Divulgaciones Matem´aticas Vol. 22, No. 2 (2021), pp. 10–22