  We study strong indispensability of minimal free resolutions of semigroup
rings. We focus on two operations, gluing and extending, used in literature to
produce more examples with a special property from the existing ones. We give a
naive condition to determine whether gluing of two semigroup rings has a
strongly indispensable minimal free resolution. As applications, we determine
extensions of $3$-generated non-symmetric, $4$-generated symmetric and pseudo
symmetric numerical semigroups as well as obtain infinitely many complete
intersection semigroups of any embedding dimension, having strongly
indispensable minimal free resolutions.
