Meronymy
Meronymy (Greek meros: ‘part’) is the relation of part to whole: hand is a meronym of arm, seed is a meronym of fruit, blade is a meronym of knife (conversely, arm is the holonym of hand, fruit is the holonym of seed, etc.). Surprisingly, not all languages seem to have an unambiguous means of translating the phrase ‘part of’ (Brown 2002: 482; Wierzbicka 1994: 488 492 disagrees), but meronymy is nevertheless often at the origin of various polysemy patterns (where a single word has more than one meaning; see 5.3 below), and an important lexical relation for that reason. Thus, according to the figures given by Brown and Witkowski, roughly one in five of the world’s languages use the same term to designate the eye (meronym) and the face (holonym) (Brown and Witkowski 1983). Similarly, slightly fewer than half of the world’s languages polysemously relate ‘hand’ and ‘arm’ as separate meanings of the same word, and 39 per cent ‘foot’ and ‘leg’ (Witkowski and Brown 1985). These figures are only estimations, but polysemy patterns based on meronymy are certainly frequent cross-linguistically. (See 11.4.1 on the semantics of body-parts in the world’s languages.)
The definition of meronymy as based on the ‘part of’ relation is not without problems. Typically, meronymy is taken to be transitive: if A is a meronym of B, and B is a meronym of C, then A is also a meronym of C. This follows what seems to be the logical structure of the part-whole relation: if A is a part of B, which is in turn a part of C, then it seems to be necessarily true that A is also part of C. The use of part of in English is often consistent with the transitivity of the meronymic relation. Thus, sequences of embedded parts and wholes, such as seed-fruit-plant, yield perfectly natural-sounding sentences highlighting the part of relation:

The transitivity of meronymy also applies for the triple cuff-sleeve-coat: a cuff is part of a sleeve, a sleeve is part of a coat, and a cuff is also part of a coat.
But the use of part of in natural language does not always respect the logically transitive nature of meronymy. Consider the relation handle-door house. While clearly we can naturally say a handle is part of a door and a door is part of a house, it seems unnatural to say that a handle is part of a house. The chain of meronymies in (11), moreover, is not only unnatural, but also false:

These facts suggest that the linguistic category part of does not have the same properties as its logical counterpart. Lyons (1977: 312) suggested that there are in fact several different types of meronymy in language. Acting on this suggestion, Iris, Litowitz and Evens (1988) isolate four different types of meronymy in English: the relation of the functional component to its whole, such as the relation between heart and body or engine and car; the relation of a segment to a preexisting whole (slice-cake); the relation of a member to a collection or an element to a set (sheep-flock); and the relation they call subset-set (fruit-food; this would normally be considered an example of hyponymy, which we discuss below). Transitivity holds for the subset and segmented wholes types of meronymy, but not for the functional part or collection-element types.
For their part, Winston, Chaffin and Herrmann (1987) propose a six-way typology, according to which part of has six possible different meanings: component-integral object meronymy (pedal-bike), member-collection (ship fleet), portion-mass (slice-pie), stuff-object (steel-car), feature-activity (paying-shopping) and place-area (Everglades-Florida). They claim that meronymy is transitive when the same type of meronymic relation is involved in all parts of the chain, as in (12), which contains the component-object type of meronymy:

Contrastingly, (11) above involves component-object meronymy in (a) and member-collection meronymy in (b); hence, transitivity fails.