If a bean is annotated by stereotype A it inherits all additional stereotypes declared in A (see 2.7.1.5). All those stereotypes are to be taken into account when computing bean name, scope, enablement, and looking for @alternative. Additional stereotypes are inherited independently of annotation @Inherit on stereotype, which only controls inheritance from super classes.

Test that shows the problem with named beans:
EXECUTE: Create stereotypes FirstStereotype, SecondStereotype and bean Bean

@Stereotype
@Named
@Target({ TYPE})
@Retention(RUNTIME)
@Documented
public @interface FirstStereotype {
}

@Stereotype
@FirstStereotype
@Target({ TYPE})
@Retention(RUNTIME)
@Documented
public @interface SecondStereotype {
}

@SecondStereotype 
public class Bean {
}

ASSERT: EL name "bean" is available.