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We answer this question in a more general form by introducing HS-stability and characterising the HS-stable involution subsemigroup generated by a subset of a given involution semigroup. We study HS-stability for the special cases of regular<jats:inline-formula><jats:alternatives><jats:tex-math>$${}^{*}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:msup><mml:mrow\/><mml:mrow><mml:mrow\/><mml:mo>\u2217<\/mml:mo><\/mml:mrow><\/mml:msup><\/mml:math><\/jats:alternatives><\/jats:inline-formula>-semigroups and commutative involution semigroups.<\/jats:p>","DOI":"10.1007\/s00233-021-10213-x","type":"journal-article","created":{"date-parts":[[2021,8,10]],"date-time":"2021-08-10T10:15:15Z","timestamp":1628590515000},"page":"395-413","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["HS-stability and complex products in involution semigroups"],"prefix":"10.1007","volume":"103","author":[{"given":"Bertalan","family":"Bodor","sequence":"first","affiliation":[]},{"given":"Erkko","family":"Lehtonen","sequence":"additional","affiliation":[]},{"given":"Thomas","family":"Quinn-Gregson","sequence":"additional","affiliation":[]},{"given":"Nikolaas","family":"Verhulst","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2021,8,10]]},"reference":[{"key":"10213_CR1","doi-asserted-by":"publisher","first-page":"93","DOI":"10.1017\/S0004972700015495","volume":"48","author":"D Easdown","year":"1993","unstructured":"Easdown, D., Munn, W.D.: On semigroups with involution. 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