What is a semantic role of "the coat" in the sentence "the coat lay on the bed"?
An entity which undergoes a change of location or possession, or whose location or possession is specified, is called a theme.
(By the way, in case this seems an odd choice of term: in Ancient Greek a thema was something "put down or placed" ... the modern sense derives from its use in the sense "proposition", something set down for discussion.)
Here's another opinion:
From Wikipedia (tidied):
In linguistics, a grammatical patient, also called the target or undergoer, is the participant of a situation upon whom an action is carried out. A patient as differentiated from a theme must undergo a change in state. A theme is denoted by a stative verb, whereas a patient is denoted by a dynamic verb. (At the very least, there is debate to this effect.) Also, patient is the name of the thematic relation with the above definition.
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The term "theme" is often used to describe the same relation as patient.
It looks like we have the usual confusion of terms.
There is a lot of confusion over terminology in this area.
One area of confusion is that theta-roles and thematic roles are often conflated, but more confusing is that it seems as though every researcher has their own take on what the roles are and what they are called. Then (as on Wikipedia...) they get listed together without any indication of whether they come for Chomsky or Fillmore or Gruber or Jackendoff or Dowty or Andrews etc. Added to that is the way that quite often the ideas overlap and differ only in detail, where the Devil lives.
If we take Chomsky to be theta-roles (related to verb arguments in a one-argument-one-role relationship), Fillmore/Gruber/Jackendoff to be thematic relations (one NP can have multiple roles), and Dowty and Andrews to be a sort of unified theory (two photo-roles - P-agent and P-patient), then we can reasonably view Jackendoff's list as a standard, based as it is on both Gruber and Fillmore.
One of the common areas of confusion is (Sod's Law strikes here...) PATIENT and THEME, and it is not uncommon to see them lumped together as 'patient/theme'.
PATIENT is reasonably stable as an entity that undergoes an action (Jackendoff's description) but THEME is all over the place, being an entity that has a position assigned to it, or an entity that changes position, or an entity wit a property assigned to it. Jackendoff gives us for verbs of motion: the moving object, for verbs of location; the thing that is located.
The thing to note here is that it is all about the verb. The verb in our sentence is lays which neither locates nor describes motion, so using Jackendoff the thematic role cannot be THEME.
That leaves us with PATIENT which makes sense. What is laying? The coat is.
Note that for thematic or theta roles, things like 'action' and 'affected' need to be seen as more heuristic than material, and we need to avoid distraction by peripheral arguments such as on the bed.