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.

• I tend to agree with you. In my opinion, stronger means less general. Saying a function is $C^\infty$ is more strong than saying a function is continuous Mar 7, 2015 at 3:22
• In general you won't find terminology like that ("stronger than") in actual [quality] mathematics publications when talking about classes of algebraic structures.
– Fizz
Mar 7, 2015 at 3:25
• Do not use neither of the two options you propose in the title. If you even have to ask, then it is evidently better to avoid this and find a clearer, more easily understandable way of saying what you want. Mar 7, 2015 at 4:25

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.

• Since this is a basic logic question you should probably add that "is stronger" means that "it implies".
– Fizz
Mar 7, 2015 at 3:29
• After thinking a bit more about this, I think it is much more a question of idiomatic/customary English usage (in Mathematics) than it is of logic. In some computer science publications, you can find expressions like "groups are a subclass of semigroups". While this is less confusing, it's probably still jarring to a mathematician. Saying that the "the class of semigroups is larger than the class of groups" is probably a more common expression in mathematics context.
– Fizz
Mar 7, 2015 at 3:49
• @RespawnedFluff The phrase wouldn't be jarring to a mathematician, though the mathematician would prefer to say "the category of groups is a subcategory of the category of semigroups."Edit: Your phrase just with class is meaningful mathematically as well, though the mathematical meaning of category is closer to the CS meaning of class, I would guess (whereas class simply means "collection").
– aes
Mar 7, 2015 at 4:24

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.

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.

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.)

• "the class of semigroups is larger than the class of groups" That sounds like a statement about cardinality, rather than containment. For example, one can just as easily say "The class of ants is larger than the class of humans" as "The class of humans is larger than the class of Americans." Mar 7, 2015 at 12:43
• @DavidRicherby: Nevertheless it is used aplenty in this sense. ex1 ex2 ex3 ex4 ex5 ex6 ex7 These are classes rather than sets, as in the case of your examples.
– Fizz
Mar 7, 2015 at 18:43
• @David Richerby: I'm curious what terminology would you use instead? "Contained in"?
– Fizz
Mar 7, 2015 at 18:44
• Yeah, I'd express that kind of idea using words like "contained in", "is a subset/subclass of", "every X is a Y", and so on. Mar 7, 2015 at 18:57
• @David Richerby: "subclass" was my first choice/idea. But I haven't found it used much in this sense [by mathematicians]; although I did find some uses ex8 ex9 ex10 they usually are at the interface between math and CS. I haven't found "contained in" used in this sense, perhaps because mathematicians are more concerned about the container not being a set.
– Fizz
Mar 7, 2015 at 19:18