Partially ordered case hierarchies

I argue that patterns of case syncretism across languages can be modelled by partially, rather than totally, ordered case hierarchies. Partially ordered case hierarchies avoid the need for postulating multiple instances of the same case while retaining case contiguity in the representation of case syncretism. I discuss data from three languages in which differential object marking is syncretic with dative and show that partially ordered hierarchies capture these and other patterns of syncretism in line with some of the conclusions reached by Harðarson (2016) and Zompì (2019) about cross-linguistic patterns of case syncretism.


Bárány
Glossa: a journal of general linguistics DOI: 10.5334/gjgl.1451 1 Introduction: Case hierarchies and syncretism Caha (2009: 18) proposes that non-accidental case syncretism can be modelled by contiguous stretches on what he calls the "case sequence", shown in (1) (see also Blake 2001: 89, 155-160;"accidental" syncretisms are the result of phonological processes, for example; see Caha 2009: Ch. 8;Zompì 2019). Caha's hypothesis is that the sequence or hierarchy in (1) correctly describes that accusative and genitive can be syncretic but nominative and genitive cannot be (Caha 2009: 11). This generalisation is called "case contiguity". (1) The case sequence (Caha 2009: 18) nominative -accusative -genitive -dative -instrumental -comitative More abstractly, the generalisation that two syncretic cases cannot be separated by another, non-syncretic case on a sequence such as (1) is an instance of the phenomenon known as *ABA, where "A" refers to the identical exponents of two cases, while "B" refers to an intervening case It has become clear, however, that certain patterns of syncretism do not straightforwardly fit the hierarchy in (1). Harðarson (2016), for example, argues for a "weak" version of case contiguity and suggests that languages can order certain cases, in particular genitive, in different ways, without losing too much of the restrictiveness of Caha's (2009; approach. A second type of modification has been proposed by Starke (2017) and is discussed by Caha (2019); Irimia (2020) (see also Caha 2009: 125-130 for an earlier, similar idea). Addressing the position of genitive in the case sequence as well, Starke (2017) proposes that rather than genitive "changing place" with respect to accusative and dative, genitive remains fixed in the sequence but is surrounded by different accusatives and datives, as shown in (2).
(2) Partial case sequence (Starke 2017: 5) nom -Sacc -Sdat -… -gen -Bacc -Bdat In (2), there are two accusatives and datives, one marked with "S" for "shifted", "structural", or "smaller" and one with "B" for "base" and "bigger" (Starke 2017: 4). Positing two accusatives allows Starke to account for syncretism of nom and Sacc as well as syncretism of Bacc and Bdat. This maintains case contiguity and avoids an *ABA pattern, but crucially only because there are two accusatives and datives. Part of the empirical motivation for positing two accusatives and datives comes from differential object marking (DOM) in Spanish (Torrego 1998;Aissen 2003;Cuervo 2003;Leonetti 2004;López 2012;Fábregas 2013;Bárány 2018; Pineda 2020) among many others), where direct objects alternate between zero-marking and being expressed with the marker a, which also serves as a dative marker. Neither is syncretic with the genitive preposition de 'of'.
Spanish DOM and dative are illustrated in (3). In (3), an inanimate direct object appears without any case marking. In Starke's (2017) terms, this means that the form of Sacc is identical to that of nom. In (3b), the definite, animate direct object is preceded by the marker a. An identical exponent marks the recipient argument in (3c). In Starke's (2017) terms, (3b,c) illustrate that Bacc is syncretic with Bdat. Yo le doy el libro a la mujer. I cl.dat give the book dat the woman 'I give the book to the woman.' Similarly, Irimia (2020) discusses languages in which DOM is syncretic with another case, for example locative in Romanian (pe) and dative in Gujarati (-ne) and varieties of Basque (-ri;Odria 2014;. For Gujarati, Irimia (2020) proposes the hierarchy or sequence in (4) with two absolutive cases (note that while Starke's hierarchy is still proposed to be universal, with languages making different choices of which cases are spelled out, this is less clear for Irimia's).
(4) Proposed case sequence for Gujarati (Irimia 2020: 41) abs 1 -erg -loc 1 -gen -abs 2(DOM) -dat -abl -… Gujarati is a split-ergative language. In the perfective aspect, subjects of transitive verbs (a) are marked ergative, while intransitive subjects (s) and transitive objects (p) are generally morphologically unmarked, or absolutive (for the labels a, s, and p, see e.g. Comrie 1989;Croft 2003). In the imperfective aspect, case-marking is neutral, meaning that a, s, and p generally appear in the absolutive (Cardona 1965: 168-170). However, in both aspects, the transitive direct object (p) can be marked with the differential object marker -ne; direct objects thus alternate between a morphologically unmarked form and DOM.
This DOM marker is syncretic with dative case (-ne). The syntactic behaviour of morphologically marked direct objects is distinct from that of true dative objects, however (see Section 2). Irimia (2020) takes this as evidence that DOM and dative objects are merely morphologically identical but represent different syntactic objects. This is the main motivation for positing a second abs in (4) which is adjacent to dat, rather than analysing the case of direct objects as dative.
In addition, Gujarati also shows syncretism of ergative, locative and instrumental case, all spelled out as -e (Mistry 2004).
A third suggestion about how to account for patterns of syncretism in light of counterevidence to (1) is Zompì's (2019) proposal that the case sequence orders classes of cases, rather than cases themselves, with respect to each other. This is shown in (5), where clausal unmarked cases are nominative and absolutive, clausal dependent cases are accusative and ergative, and inherent cases are all others (apart from genitive). In each class in (5), cases need not be (totally) ordered with respect to each other. The hierarchy in (5) underlies the generalisation in (6).

The proposal in a nutshell
In this paper, I also propose an analysis of patterns of syncretism that does not presuppose a total order on a case hierarchy but which nevertheless retains case contiguity. I suggest that rather than conceiving of a case sequence as a total order, it can be thought of as a partially ordered set (or poset) such that not all elements of the case "hierarchy" need to be ordered with respect to each other (Figure 1; see also Bárány 2016; Graf 2019).   (Partee, Meulen & Wall 1990: 278).
Partially ordered sets can be used to account for case syncretisms. Rather than duplicating cases on a totally ordered hierarchy, a hierarchy can be "split" into branches. The elements of two distinct branches are incomparable to, that is not ordered with respect to, each other, but they are comparable to, that is ordered with respect to, other elements not on those branches. Again, {A, Z}, {A, B}, and {A, B, C, Y, Z} in Figure 1 illustrate this. This allows for maintaining contiguity between syncretic cases by not ordering them with respect to cases that intervene on totally ordered hierarchies. More concretely, rather than ordering Gujarati's cases as in (4), repeated here, they can be ordered as in (7) to account for the actual case paradigm of Gujarati, maintaining strict contiguity of the empty set ∅ and abs on the one hand, and abs and dat on the other.
(4) Proposed case sequence for Gujarati (Irimia 2020: 41;repeated) abs 1 -erg -loc 1 -gen -abs 2(DOM) -dat -abl -… (7) Partially ordered Gujarati case sequence (to be revised) The hierarchies proposed in this paper are an abstract representation of cases and their underlying features which are related to surface forms by spell-out rules. In this system, syncretism arises through underspecification, when one vocabulary item can spell out several sets of features. Underspecification can be part of the system as a whole (e.g. two forms being syncretic in all contexts) or the result of the application of impoverishment rules. These two sources of syncretism are not redundant, however, as only syncretism through impoverishment leads to alternations, namely differential object marking in the context of this paper.
This approach naturally models a scenario in which syntactic objects with different morphological forms show identical syntactic behaviour, because they are of the same underlying type. I will argue that this is the case for direct objects in Gujarati, Basque, and Spanish, motivating that they are associated with a single level on a hierarchy.
This contrasts with Starke's (2017) and Irimia's (2020) hierarchies. These capture patterns of syncretism and maintain case contiguity, but at the price of positing structural (or dependent) cases in between inherent (or oblique) ones, resulting in hierarchies that do not conform to Zompì's (2019) universal containment. At least Starke's (2017) hierarchy also seems to be intended as a (universal) underlying representation (Starke 2017: 1, 4, fn. 2), the levels of which are spelled out differently in different languages (Starke 2017: 5). If so, I believe the data discussed in this paper provide strong evidence against positing two distinct underlying representations of acc. But even if one takes Starke's (2017) and Irimia's (2020) hierarchies to be surface representations, they do not conform to Zompì's (2019) universal containment because they interleave structural and inherent cases.
The partially ordered hierarchies described below fare better in this respect. First, two distinct morphological forms can be associated with one underlying syntactic object. Second, this approach captures the patterns of syncretism of DOM and dative case in Gujarati, Basque, and Spanish just like approaches based on total orders (see e.g. Bárány 2018), but it also captures erg=loc syncretism in Gujarati, while maintaining case contiguity (avoiding *ABA patterns) and universal containment. Finally, by "branching off" certain cases, genitives can be included in partially ordered case hierarchies in a systematic way. While the hierarchies shown below are compatible with Zompì's (2019) generalisation in (5), they can vary from language to language, based on the respective case paradigm of a language, as in Harðarson (2016).
In the remainder of this paper, I will argue in favour of partially ordered case hierarchies in more detail. The outline of the argument is as follows. The data in Section 2 indicate that morphologically unmarked and DOM objects show identical syntactic behaviour and that they behave differently from other syntactic relations, such as subjects and indirect objects, even if they are syncretic with them. I take this to mean that both unmarked and DOM objects are direct objects. In Section 3, I therefore analyse morphologically unmarked and DOM objects as representing the same syntactic relation and being associated with the same underlying Case which can be spelled out in different ways. Finally, I show that partially ordered case hierarchies successfully account for these patterns while conforming to Zompì's (2019) universal containment.

Data
In a number of languages DOM has the same morphological expression as dative case (see Manzini & Franco 2016;Bárány 2018;Irimia & Pineda 2019). In this section, I discuss Gujarati, Basque and Spanish (in this order). For each language, my aim is to demonstrate that objects in these languages show similar syntactic properties.
I argue that, first, both DOM objects and morphologically unmarked objects show identical syntactic behaviour, independently of their morphological form and, second, that the behaviour of these objects is typical of direct objects and distinguishes them from other syntactic relations such as subjects and indirect objects, even when syncretic.
I will argue in Section 3 that these characteristics motivate identifying both morphologically unmarked and DOM direct objects as relations typically encoded by abs in ergative-absolutive languages and acc in nominative-accusative languages. Therefore, they represent the spell-out of abs or acc on case hierarchies, respectively.
Gujarati also has a complex agreement system (Comrie 1984;Mistry 1997;Grosz & Patel-Grosz 2014). Addressing it in detail goes beyond the scope of the present paper, but agreement is relevant in two respects. First, it can distinguish the behaviour of different arguments marked with -ne. In contrast to Hindi-Urdu, where only morphologically unmarked noun phrases can control agreement (Saksena 1981;Comrie 1984;Mohanan 1990;1994;Bhatt 2005;Butt 2006), morphologically marked direct objects can control agreement in Gujarati, as shown in (8). Raju-e Svati-ne jo-y-i.
Examples (8a,b) are in the perfective aspect, with ergative subjects (-e). In (8a), the direct object is morphologically unmarked and controls gender agreement. The verb references the

Case Suffix
Absolutive object's masculine gender rather than the subject's feminine gender. In (8b), the proper name object Svati-ne is morphologically marked but still controls gender agreement. Ergative noun phrases never control agreement (Comrie 1984;Mistry 1997).
The suffix -ne, glossed as dom in (8), is also the exponent of dative. In constructions with dative arguments, as in (9) and (10), the dative cannot control agreement, however, independently of aspect. (9) Gujarati (Mistry 2004: 7) a.
The second relevant aspect of agreement is that it also distinguishes between subjects and objects. Grosz & Patel-Grosz (2014: 222) argue that Kutchi Gujarati (and potentially other varieties of Gujarati, too) have two φ-probes, one on T and a second, lower one on v or Asp. In (11), the auxiliary spells out agreement with the subject while the participle spells out agreement with the object. Note that it does not matter whether the object shows DOM or not, it triggers the same type of agreement, suggesting that both types of objects are in a relation with v. Indeed, Grosz & Patel-Grosz (2014: 241) conclude that DOM "appears to be connected to abstract (accusative) case licensing by v".

(11)
Kutchi Gujarati (Grosz & Patel-Grosz 2014: 222, Pritty Patel-Grosz, p.c.) Hu chokra(-ne) jo-y-a ha-is. I boys-dom see-pfv-pl aux-fut.1sg 'I will have seen the boys.' In sum, Kutchi Gujarati agreement patterns show that direct objects pattern together, independently of DOM, and that both types pattern differently from indirect objects as well as subjects (see Grosz & Patel-Grosz 2014 for more detailed discussion). This is strongly indicative of a distinct type of syntactic relation (direct object) which, independently of its morphology, shows characteristic syntactic behaviour that crucially differs from that of other syntactic relations, even if their expression is syncretic.  Odria (2014; shows that in some southwestern varieties of Basque, direct objects are either morphologically unmarked (abs) or appear with the case marker -ri. Odria argues that -ri is the syncretic expression of DOM and dative case based on the different syntactic behaviour of direct and indirect objects with -ri.

Basque
One such difference is an argument's ability to control depictive secondary predicates. Odria (2014) points out that indirect objects cannot control depictive secondary predicates while direct objects can. In (12a), the ergative subject ni-k 'I-erg' and the absolutive object umea 'child' can control the secondary predicate pozik 'happy', but the dative amona-ri 'grandmotherdat' cannot. In (12b), in contrast, the direct object zu-ri 'you-dom' can control the secondary predicate. This difference thus cannot be due to the morphology of -ri.
Ni-k i amona-ri j umea k pozik i/*j/k eraman I-erg grandmother-dat child.abs happy carry Other arguments for treating -ri as DOM rather than dative case come from its sensitivity to person in certain varieties and its behaviour in ditransitive constructions. Odria (2014: 297-300) shows that in Zumaia Basque first and second person direct objects are marked with -ri, (13), but third person objects are not. In other varieties, third person objects can be marked with -ri as well, as long as they are human and definite. These patterns are typical of DOM, but not of datives. Ni-k zu-ri ekarri d-i-zu-t. I-erg you-dom bring tm-(root)-df-2.dat-1sg.erg 'I have brought you.' In ditransitives, where the recipient argument is marked with dative -ri, however, the same direct objects that are marked in (13) lose DOM, as shown in (14). (14) Southwestern Basque (Odria 2014: 304) a. Traidori-k ni etsaia-ri saldu n-a-u-te. traitor-erg I.abs enemy-dat sell 1sg.abs-ep-root-3pl 'The traitors have sold me to the enemy.' b.
Traidori-k zu etsaia-ri saldu z-a-itx-u-zte. traitor-erg you.abs enemy-dat sell 2.abs-ep-root-3pl 'The traitors have sold you to the enemy.' This behaviour of DOM objects is also found in Hindi and Spanish (Bhatt & Anagnostopoulou 1996;Ormazabal & Romero 2013;Odria 2014). Crucially, in all of these languages, it is the direct object that loses its case marker, never the indirect object. This distinguishes the direct from the indirect object syntactically and again shows that direct objects can be morphologically marked or unmarked. In addition, in varieties of Spanish, Hindi and Basque, DOM is optional in ditransitive constructions even on direct objects that would otherwise require DOM. I will not provide an analysis of this phenomenon here, but it could be captured by Richard's (2010) "distinctness".
In sum, the data shown here suggest that DOM objects in Basque do not pattern syntactically with dative objects but rather with unmarked absolutive objects (Odria 2014: 310 with Odria's (2014) assessment that both absolutive and DOM objects are syntactically canonical direct objects.
Thus Basque shows the same characteristic properties as Gujarati with respect to DOM and dative case. Morphologically unmarked and DOM objects that behave alike syntactically. They are direct objects whose syntactic behaviour can be distinguished from that of other syntactic relations, for example with respect to person sensitivity and being able to control secondary predicates.

Spanish
It has been argued that the marker a is the syncretic spell-out of distinct underlying structures representing DOM and dative objects, respectively, rather than a single type of syntactic structure (see e.g. c. *Recipient passive of a ditransitive *La mujer fue dada el libro. the woman was given.f the book intended: 'The woman was given a book.' In addition to Bárány's (2018) data in (15), (16) shows that a theme passive of a ditransitive is also possible (in contrast to the recipient passive in (15c)). (16) is modeled after the active examples (52) and (53) in Ormazabal & Romero (2007) in which DOM is obligatory for the direct object a los sospechosos.

(16)
Spanish (Víctor Acedo-Matellán, p.c.), Theme passive of a ditransitive Los sospechosos fueron entregados a la policía. the suspects were handed.over.m.pl dat the police 'The suspects were handed over to the police.
Passives therefore distinguish DOM objects from dative objects and group morphologically marked and unmarked direct objects together. Demonte (1987;); Odria (2014; 2019); Bárány (2018); López (2018) provide further evidence that morphologically unmarked and DOM objects differ from dat objects. Demonte (1987;Odria (2014; show that direct objects (with or without DOM) can control secondary predicates (like in Basque), and López (2018) shows that dative case is retained in nominalisations while both unmarked and DOM direct objects are marked by the genitive de. Finally, Bárány (2018) argues that passivisation and controlling secondary predication are not tied to thematic roles: the verb armar 'to arm' takes a recipient direct object which behaves just like other direct objects in that it can be morphologically unmarked or appear with DOM ( (17)), control secondary predication ((18a)), and be passivised ( (19)), in contrast to dative recipients in ditransitives.
Spanish (Bárány 2018: 28) Los soldados fueron armados por el gobierno con pistolas. the soldiers were armed.pl.m by the government with pistols 'The soldiers were armed by the government with pistols'.
Like Gujarati and Basque, Spanish shows characteristic properties of languages with DOM/dat syncretism. The language has DOM and morphologically unmarked and DOM objects which behave alike syntactically, like direct objects. These direct objects behave differently from other, syncretic, syntactic relations.

Interim summary
In this section, my aim was to show that Gujarati, Basque and Spanish all show similar properties, characteristic of languages with DOM and dative syncretism. All have a syntactic relation that corresponds to direct objects which can be morphologically unmarked or be spelled out as DOM and in all three languages these objects show strikingly different syntactic behavior from other syntactic relations such as subjects or objects even when syncretic. The properties I discussed are summarised in Table 2. I am not aware of any evidence that would suggest that these direct objects differ in their syntax from typical absolutive or accusative objects in ergative-absolutive and nominative-accusative  languages. 1 This view is widely held in the literature. Starke (2017); Bárány (2018) refer to both morphologically unmarked and DOM objects as accusative (small and big in Starke 2017), while Odria (2014; and Irimia (2020) characterise both as being like absolutive direct objects.
In the following section, I develop an analysis that retains case contiguity and case containment while capturing both the identical syntactic behaviour of morphologically unmarked and DOM objects and their patterns of syncretism.

Analysis
In this section, I show how partially ordered case hierarchies can capture the patterns of syncretism reviewed in Section 2 while maintaining case contiguity and conforming to Zompì's (2019) universal containment hypothesis.
The hierarchies proposed here represent underlying sets of case features, from which patterns of syncretism are derived at spell-out. This means that a single case, for example accusative, can underlie two distinct morphological forms. The distinct exponents of morphologically unmarked and DOM objects are a result of impoverishment rules which modify the underlying features before these are spelled out. As I will show, this allows the hierarchies to conform to universal containment and case contiguity and to naturally account for the syntactic behaviour of morphologically unmarked and DOM objects.
I first lay out how partially ordered case hierarchies can account for the data just discussed (Section 3.1) before arguing in more detail that both DOM objects and morphologically unmarked objects represent the same type of object (Section 3.2).

Modelling partially ordered case hierarchies
The starting point for my proposal is case decomposition. Rather than treating "nominative", "ergative" etc. as atomic, it has often been suggested that these categories are bundles of features. Many analyses of this type, for example those by Bierwisch (1967) . This is true both when each case is spelled out by a distinct vocabulary item and when two or more cases are syncretic. Consider the two sets of spell-out rules in (21) referring to the case features in (20). In the absence of another spell-out rule referring to dat in (20), (21a) describes a situation in which the marker -er would be inserted for both gen and dat. This is due to DM's subset principle, which states that a vocabulary item, for example -er in (21a), must have a subset of the features of the terminal node it is inserted in (see e.g. Halle & Marantz 1993: 121-122). The single marker -er in (21a) thus syncretically spells out gen and dat. Note that no impoverishment is necessary here; the set of vocabulary items is merely underspecified. (21b), in contrast, describes two distinct markers for gen and dat.  (20), gives rise to a hierarchy totally ordered by the subset relation ⊂. As mentioned in Section 1, this order has been argued to be empirically inadequate: taking Spanish as an example, where a spells out both DOM and dative but de spells out genitive, the syncretism of DOM and dative violates case contiguity if DOM represents accusative.
Instead of giving up cumulative classification and ordering by ⊂, or adding several instances of accusative and dative, I propose to give up the total ordering of (20), within limits. To restrict the expressive power of the partially ordered hierarchies, I will maintain the general idea behind Zompì's (2019) (5), namely that unmarked cases (nom, abs) are ordered before dependent cases (acc, erg), which are in turn ordered before inherent cases. More specifically, my hierarchies implement Zompì's (2019) generalisation as stated in (22). This allows for unmarked cases to be subsets of dependent cases, but crucially also for (some) dependent cases to be incomparable to (some) inherent cases. (22) Partially ordered hierarchies and universal containment Unmarked cases are subsets of dependent cases. No inherent case is a subset of a dependent case.
As for possible DOM syncretisms, this system correctly accounts for syncretism with dat (as discussed in this paper) but also with ins (as in Kolyma Yukaghir, Maslova 2003;Keine 2010) or gen (as in Finnish, Kiparsky 2001;Vainikka & Brattico 2014;Poole 2015). In principle, like on Zompì's (2019) approach, DOM could be syncretic with any inherent case, although crosslinguistically dat seems to be the most frequent choice (Bossong 1991). By maintaining case contiguity the system rules out syncretism of an inherent case with an unmarked case, unless a dependent case is also syncretic (*ABA configurations). In this respect, it is at least as powerful as Starke's (2017) and Irimia's (2020)  In line with the data in Section 2 and the subsequent discussion, I argue that both morphologically unmarked and DOM objects represent direct objects which typically correspond to abs or acc in languages without differential marking. Recall also that the cases on (23) are underlying representations, the features of which can be subject to impoverishment before spell-out. To avoid such violations, the present proposal provides the solution that a learner confronted with a language exhibiting a non-accidental syncretism of this sort would posit a branching, partially ordered hierarchy, where gen and acc are incomparable. This way, non-accidental syncretism through impoverishment would always involve deleting a single case feature and targeting adjacent sets (restricting its power). If a language were to exhibit accidental syncretism of unmarked and oblique cases, impoverishment could derive this, but with more complex rules. For Spanish, the spell-out of nom, acc, gen, and dat on the hierarchy in (23) can be analysed as follows. acc and dat are now contiguous, without positing two instances of each case, as gen does not intervene between them. By assigning the cases the features shown in (23), syncretism of acc and dat can be straightforwardly implemented in spell-out rules.
The spell-out rules for de, the genitive, and a, the syncretic form of DOM and dative, are shown in (24).
(24) Vocabulary items for Spanish case markers based a {A, B, Z} ↔ de b.
{A, B} ↔ a By the subset principle, a in (24b) can be inserted for both acc with the feature set {A, B} and dat with the feature set {A, B, C}. {A, B} is of course also a subset of gen with {A, B, Z}, but there is a more specific marker, de, which takes precedence over a. By giving up the total order of case sequence, this approach to Spanish acc, dat, and gen retains case contiguity, that is it does not derive non-linear case syncretisms (*ABA patterns).
The same approach can account for the more complex patterns in Gujarati and (varieties of) Basque as well. Recall that Gujarati is split-ergative, while Basque is ergative. Both have morphologically unmarked abs direct objects as well as morphologically marked DOM that is syncretic with dative. Having a morphologically marked direct object case in ergative languages is not very common; such case systems are referred to as "tripartite" because a, s, and p all have distinct morphological forms (see e.g . Deal 2010;Comrie 2013;Baker 2015). Comrie (2013) refers to Hindi-Urdu as a tripartite language because, like Gujarati and Basque, it has DOM in ergative contexts such that both the transitive subject and the object have morphological casemarking. In this respect, Gujarati and Basque differ from Spanish: Spanish only spells out one dependent case (in Zompì's terminology), acc. Gujarati and Basque can spell out two dependent cases, erg and acc. This motivates the inclusion of abs, erg, and acc in the hierarchy in (25) (see Section 3.2) for further discussion).
In Gujarati, ergative and locative are also syncretic. A partially ordered Gujarati case hierarchy is shown in (25), annotated with the spell-out of each marker, its features and Zompì's (2019) categories.
(25) Partially ordered Gujarati case hierarchy and universal containment In (25), not all cases are comparable, or ordered with respect to each other by ⊂, hence it represents a partial order. erg and loc are incomparable to dat, while both groups are comparable to abs and gen. Crucially, erg, a dependent case, is ordered before loc, an inherent one, and dat does not intervene between acc and erg, as erg and dat are incomparable. Therefore, no inherent case precedes a dependent case and no dependent case precedes an unmarked case.
The vocabulary items spelling out these cases are shown in (26).  The Basque syncretism of DOM and dative can be captured in essentially the same way as the Gujarati one: acc ⊂ dat, but both are incomparable to erg. This makes it possible for a single vocabulary item -ri to spell out both DOM and dat.
The present implementation of partially ordered case sequences is similar in some respects to Harðarson's (2016) proposal that case contiguity is weak, Zompì's (2019) proposal that classes of cases, rather than individual cases, are comparable, as well as Bárány's (2016) and Graf's (2019) partial orders. Harðarson's (2016) hierarchies represent total orders, which may, however, differ across languages. Zompì's (2019) hierarchy in (5) is universal, but refers to classes of mutually unordered cases. Similarly, Graf (2019) posits different levels on which cases are incomparable.
Partially ordered hierarchies capture the benefits of both Harðarson's (2016) and Zompì's (2019) proposals. gen can appear to be in different orders with respect to acc and dat by being unordered relative to dat, as in (27) (repeated from (23)). Both types can be captured by the same sequence.
An earlier version of Zompì's proposal, based on the data in Zompì (2017), is characterised by Graf (2019: 19) as being "both too permissive and too restrictive". Graf (2019: 20-21) instead proposes a partial order with groups of incomparable cases organised on three levels, crucially not ordering acc, erg, gen, and loc with respect to each other (note that Graf includes nom, abs, as well as acc and erg). These layers are shown in (29). Graf (2019) proposes the generalisation in (30), banning certain "parallel" syncretisms between the second and third layers of (29). 3  Graf (2019: 22) correctly points out that (29) and (30) restrict the power of partial orders. My proposed hierarchies are as permissive as Graf's layers; they could also be supplemented by a ban such as (30).
Finally, another potential advantage of the present proposal lies in the mapping of cases on the hierarchy to their morphological expression. In the case of Spanish, a single acc case underlies two different morphological forms of direct objects, in contrast to Starke's (2017) two. This is advantageous, because, as Zompì (2019: 15) points out, Starke's (2017) hierarchies both overand undergenerate. In addition, being associated with a single underlying case makes sense in light of the syntactic behaviour of direct objects with and without DOM in Spanish.
The situation is somewhat different in Basque and Gujarati, for which I argued that both abs and acc can be motivated on the hierarchies. I turn to this point now.

Direct objects, accusative, and absolutive
In this section, I return to the question of the syntactic relations and the underlying case of the objects discussed in Section 2. I will argue that the relevant objects, be they morphologically marked or not, all represent direct objects and are the (potentially modified) spell-out of an underlying acc.
The overall logic of my argument is as follows (I am grateful to two reviewers for pushing me to clarify this reasoning). First, if certain syntactic arguments show identical behaviour, it is reasonable to treat them as representing one syntactic relation. Second, if this syntactic behaviour is that of typical direct objects, it is reasonable to assume that the arguments represent that grammatical function and have abstract Case associated with direct objects (acc). Therefore, third, if morphologically unmarked direct objects and DOM show the same syntactic behaviour, they can be classified as the same syntactic relation and as having the same abstract Case -even if DOM objects are syncretic with another case, namely dat. I will address these points in turn.
I have interpreted the data in Section 2 as evidence that subjects (morphologically unmarked or ergative) can be distinguished from direct objects (morphologically unmarked or DOM) and indirect objects (dative) in Gujarati, Basque, and Spanish, based on their syntactic behaviour. Unmarked and DOM objects are able to control agreement in Gujarati, they control depictive secondary predicates and show sensitivity to person and animacy in Basque, and they can be passivised and control depictive secondary predicates in Spanish ( Table 2). Recall also that, at least in Spanish, these properties are independent of the thematic role of the object (cf. (17), (18), (19)).
This behaviour is typical of direct objects (although there is some cross-linguistic variation; see Haspelmath 2018 on difficulties of comparison of this sort). For example, passives generally promote a direct (or primary) object to subject (Oehrle 1976;Perlmutter & Postal 1983); in languages with object agreement, direct (or primary) objects are always potential agreement controllers in monotransitive sentences (Faltz 1978;Bárány to appear). The situation with secondary predicates seems to vary more across languages. While Bárány (2018) shows that in Spanish, recipient direct objects (of the verb armar 'to arm') can control secondary predicates, this is not true of goal or recipient arguments in English and German in many cases (Williams 1980;Maling 2001). Nevertheless, it is clear from Odria's (2014) data that DOM objects, even though they are syncretic with dative, behave like morphologically unmarked objects, and not like datives. Thus, depictive secondary predicates do work as a diagnostic for whether morphologically unmarked and DOM objects behave alike, as typical direct objects.
What about the (abstract) Case of these objects? In nominative-accusative languages, acc is taken to be the abstract Case associated with direct objects ("objective" in Chomsky 1981: 170; accusative assigned by v in Chomsky 2001: 6). Legate (2008), among many others, also adopts this idea and adds that v assigns abstract Case to the same class of object in ergative-absolutive languages of the type discussed in this paper.
Linking this to the case hierarchies proposed here, I assume that v assigns to the direct object the set of features associated with acc on the hierarchy. In nom-acc languages with morphological case, but without DOM, this is spelled out as acc (e.g. in German or Hungarian). In erg-acc languages with morphological case, but without DOM, this is spelled out as abs, i.e. as an unmarked case (Legate's 2008 abs=def(ault) languages). In languages with DOM, be they nom-acc or erg-acc, the underlying features of acc can be spelled out as DOM or they can undergo impoverishment and be spelled out differently (Keine & Müller 2015), for example as unmarked case in Gujarati, Basque, and Spanish.
The association of two distinct morphological forms with one underlying Case is compatible with the identical syntactic behaviour of morphologically unmarked and DOM objects and the distinct behaviour of true dative objects, which are assigned a different set of case features.
In sum, I argued that, first, morphologically unmarked and DOM objects show the same syntactic behaviour and that this supports treating them as the same type of syntactic relation. Second, this syntactic relation is that of typical direct objects, which are generally assumed to be assigned acc (or abs) in frameworks adopting abstract Case. Third, I showed that the behavior of these objects distinguishes them from that of other syntactic relations, such as indirect objects, even if these are syncretic with DOM objects.

Summary
Summarising this section and the proposal, I have argued that Harðarson's (2016) and Zompì's (2019) arguments about case contiguity can be implemented by adopting partially ordered case hierarchies. By restricting partial orders to respect Zompì's (2019) generalisations about classes of cases in (5) and syncretism in (6) as stated in (22), this proposal accounts for at least the languages covered by those approaches and, crucially, avoids Starke's (2017) multiple instances of the "same" dependent case. The analysis presented here is similar to Graf's (2019), but corresponds to Zompì's (2019) universal containment more directly. The hierarchies are partially ordered by a subset relation (⊂). Syncretism can be modelled in the same way as on totally ordered hierarchies.

Open questions
Before concluding, I will briefly address open questions. First, like in Harðarson (2016); Starke (2017); Zompì (2019), the placement of genitive with respect to other cases is not generalised in this paper. Harðarson (2016) suggests that languages can reorder dative and genitive in particular and Zompì (2019) notes that the genitive does not fit any of the three classes of cases he identifies. Starke's (2017) multiple accusatives and datives are placed on either side of genitive. My proposal does not shed any new light on why genitive behaves the way it does, but it allows for a simpler modeling of "alternating orders" of accusative, dative, and genitive by removing the need for total orders among these cases. The same stretch of a partially ordered hierarchy, as in (27), could underlie different patterns of syncretism across languages, while still only using one instance of each case.
Second, the features used in the partially ordered hierarchies presented here are abstract and meaningless. While this is sufficient for the purposes of this paper, I proposed particular hierarchies fitted to the three languages under discussion. But how are they restricted more generally? On the one hand, they conform to Zompì's (2019)'s (5) (see also (22)). This restricts expressive power of the hierarchies under discussion.
On the other hand, Harðarson (2016Harðarson ( : 1339Harðarson ( -1342 argues that if certain features are dependent on others, such that a particular feature [oblique] depends on a feature [dependent], hierarchies in which "oblique" cases precede "dependent" ones can be ruled out (see also Smith et al. 2019). An approach along these lines is also in line with Zompì's (2019) observations that dependent cases must precede (be subsets of) oblique cases on case hierarchies as well.
The order of features could reflect their frequency, the way they are acquired, and in turn their morphological expression. Morphologically unmarked cases, nominative and absolutive, as intransitive and transitive subjects or intransitive subjects and transitive objects, respectively, are frequent and have a wide distribution. "Dependent" cases, accusative and ergative, are by the definition the first that are distinguished from morphologically unmarked cases (see e.g. Yip in the domain of case explicitly to Dresher's (2009) work on phonology. This suggests that general mechanisms of acquiring featural distinctions could lead to "emergent" case hierarchies in line with Zompì's (2019) generalisations.
Third, Caha (2017) discusses Blansitt's (1988) generalisation: dat-all-loc case can only show contiguous syncretisms, that is dat and loc are never syncretic to the exclusion of all. However, in Tigrinya the markers corresponding to these cases are ne, nab (< ne ab) and ab (Blansitt 1988: 174), indicating that all contains (features of) both dat and loc. This is a problem for approaches assuming containment and a total order of cases. Caha (2017) proposes that all has features of both dat and loc (see also Bobaljik & Sauerland 2018). With partial orders, the same result can be achieved if dat and loc are incomparable to each other but comparable to all (and other cases, such as acc). This derives the syncretism and retains the benefits of contiguity and containment, but potentially overgenerates, suggesting that some additional restriction is necessary.
Finally, I have not considered whether the feature-based analysis of partially ordered case hierarchies presented here could equally well be expressed in a framework where each case feature represents a nominal projection, such as Nanosyntax. I refer the reader to the comparison of case hierarchies in DM and Nanosyntax in Zompì (2019).

Conclusions
In this paper, I proposed that case syncretism can be modelled by case hierarchies which are not totally, but only partially ordered. In analyzing patterns of syncretism involving DOM and dative case in Spanish, Gujarati, and Basque, I showed that this approach need not posit multiple instances of single cases on a hierarchy and, importantly, that the resulting partially ordered hierarchies maintain case contiguity and universal containment, in line with suggestions by Harðarson (2016) and Zompì (2019).