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We introduce the notion of a ribbon-clasp surface-link, which is ageneralization of a ribbon surface-link. We generalize the notion of a normalform on embedded surface-links to the case of immersed surface-links and provethat any (immersed) surface-link can be described in a normal form. It is knownthat an embedded surface-link is a ribbon surface-link if and only if it can bedescribed in a symmetric normal form. We prove that an (immersed) surface-linkis a ribbon-clasp surface-link if and only if it can be described in asymmetric normal form. We also introduce the notion of a ribbon-clasp normalform, which is a simpler version of a symmetric normal form.

Ribbon-clasp surface-links and normal forms of singular surface-links