The t-class semigroup of an integral domain is the semigroup of the isomorphy classes of the t-ideals with the operation induced by ideal t-multiplication. This paper investigates ring-theoretic properties of an integral domain that reflect reciprocally in the Clifford or Boolean property of its t-class semigroup. Contexts (including Lipman and Sally-Vasconcelos stability) that suit best t-multiplication are studied in an attempt to generalize well-known developments on class semigroups. We prove that a Prüfer v-multiplication domain (PVMD) is of Krull type (in the sense of Griffin [M. Griffin, Rings of Krull type, J. reine angew. Math. 229 (1968), 1–27]) if and only if its t-class semigroup is Clifford. This extends Bazzoni and Salce's results on valuation domains [S. Bazzoni and L. Salce, Groups in the class semigroups of valuation domains, Israel J. Math. 95 (1996), 135–155] and Prüfer domains [S. Bazzoni, Class semigroup of Prüfer domains, J. Algebra 184 (1996), 613–631], [S. Bazzoni, Idempotents of the class semigroup of a Pruüfer domain of finite character, Lect. Notes. Pure Appl. Math., Dekker, 201 (1998), 79–89], [S. Bazzoni, Groups in the class semigroup of a Prüfer domain of finite character, Comm. Algebra 28 (11) (2000), 5157–5167], [S. Bazzoni, Clifford regular domains, J. Algebra 238 (2001), 703–722].