Gluing semigroups and strongly indispensable free resolutions

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.