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What is the name of this type of relation between elements?

All jacuzzis are hot tubs, but not all hot tubs are jacuzzis.

All words are morphemes, but not all morphemes are words.

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One should differentiate between, on the one hand, the (logical) relation between the actual objects referred to by the words, and, on the other hand, the (linguistic) relation between the words themselves (i.e. the signs that denote these objects).

The logical relation of the entities involved is, as suggested by sumelic's comment, the proper subset relation:
The set of jacuzzis is a proper subset of the set of hot tubs; and the set of hot tubs is a proper superset of the set of jacuzzis.
In mathematical terms, a set A is a proper subset of B (A ⊂ B) if and only if every every element of A is also an element of B (A ⊆ B), and A is not identical to B (A ≠ B), so there is at least one element of B which is not an element of A.
This means that everything which is a jacuzzi is also a hot tub, but a jacuzzi is not the same thing as a hot tub, so there are hot tubs which are not jacuzzis.

The linguistic relation between the lexical items resembles (!) one of hypernymy and hyponymy:
The word jacuzzi would be a hyponym to the word hot tub and the word hot tub would be a hypernym to the word jacuzzi - if one regarded "hot tub" as a word, which is not entirely accurate because it actually consists of two words, so this terminology which is defined on words doesn't apply here that nicely.
A more straightforward example would be something like cat being a hyponym to animal and animal being a hypernym to cat (every cat is an animal, but not every animal is a cat).

As curiousdannii pointed out, the statement about morphemes and words is linguistically not corret, but if it were, we could use the same description.

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  • Thank you! I've always focused on syntax so semantics never occurred to me. – user15493 Mar 18 '17 at 1:55

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