How should the term "stronger" be used? Example: groups stronger than semigroups or semigroups stronger than groups? Everyone is well-known that every group is a semigroup. Then I should say that "group is stronger than semigroup" or "semigroup is stronger than group". Someone told me that stronger means more general. But, I think that "group is stronger than semigroup" since it has the strong condition while a semigroup is more general than group.
 A: The condition "being a group" is stronger than the condition "being a semigroup."
A theorem whose hypothesis requires a semigroup is stronger than a theorem whose hypothesis requires a group.
A theorem whose conclusion tells you that some object is a group is stronger than a theorem whose conclusion tells you that the object is a semigroup.
Just saying "group is stronger than semigroup" or vice versa doesn't seem to mean much.
A: I think it's confusing saying something like "a group is stronger than a semigroup", but it'll be widely understood if you say that a group satisfies stronger assumptions than a semigroup, because you're conveying that a group satisfies the same conditions as that of a semigroup, but with additional/special properties.
A: If you want to compare the classes directly it's better to say "the class of semigroups is larger than the class of groups". This is much less likely to be misunderstood by a native English, Mathematics audience.
The phrase "stronger than" is hardly ever used to compare classes, sets etc., instead being usually reserved for comparing logic conditions [predicates] as Kundor's answer in particular emphasizes. In this logic context, "stronger than" means "it implies".
So, in summary, the "condition of being X is stronger than the condition of being Y" is equivalent with saying that "the class of Y is larger than the class of X". Because of this reversal of X and Y in these two sentences, if you try to use "is stronger" as a synonym for "is larger" in the latter sentence, then you will confuse your audience basically because "is stronger" will lead them to believe you're talking about conditions, while you're talking about classes.
EDIT: after conversation with David Richerby, "X is a [proper] subclass of Y" is also a reasonably common choice of words and is more precise. My concerns about this phrase not being used [much] outside computer science turned out unfounded. I even found an "informal" definition for it in Tourlakis, vol.2, p. 139. (And of course there's nothing wrong with just sticking with "every X is Y", but the OP knew that phrase.)
A: I think the best way to think about this is to consider the notion of being 'stronger' as a relation that partially orders assertions. Ie, groups are not stronger than semigroups (or vice versa), but the assertion that an object is a group is a stronger assertion than the assertion that an object is a semigroup.
This is similar to the other answers, but I feel like a linguistic question demands a linguistic answer. The term assertion is critical.
