http://www.ut.ee/Ural/kynnap/kpls.html
'The "fans" of the population from the Iberian and the Ukrainian refuges, in the Lapps' area of departure on the North Sea Land (see Fig. 2 and especially Fig. 4) towards Scandinavia, partly overlap each other (see Fig. 1). Therefore Wiik supposes some substratum of the Basque type in the then Lapp language admitting, though, that he cannot concretely show it. Such substratum is still worth looking for in the present-day Lapp languages by way of comparison with a single preserved Basque type language namely Modern Basque. Michel Morvan has shown (Morvan 199?: 36) that already about the middle of the 19th c. Louis-Lucien Bonaparte indicated the possibility of connection of the Basque and Lapp (and Hungarian) plural marker -k (Bonaparte 1862). The possibility of connection of the plural marker -k in Basque and Lapp substantives is noted also by Morvan himself, cf. e.g. Basque guk Euskaldunok "nous les Basques" (Morvan 199?: 192193). However, from the aspect of the Uralic language group I regard the plural marker of the Lapp substantives rather as a detached phenomenon of foreign origin why not the Basque substratum than some common-Uralic suffix. Besides Lapp languages the plural marker -k of substantives occurs only in Hungarian but even this is exceptional. Angela Marcantonio notes about the Hungarian plural marker -k (Marcantonio 2002: 234235), "Unlike most U[ralic] languages, Hungarian has a different Plural ending, used both for nouns ... and for verbs: the ending -k. A Plural -k is found also in Lapp, although this is generally considered as deriving from *-t ... [...] The origin of *-k is disputed. Some researchers believe that it derives from a derivational suffix *-kkV ... This explanation looks a bit far-fetched. [...] Aalto ... considers the possibility of connecting -k with the Samoyed co-affixal element *-k(ø)- ..., as well with the Tungus, Turkic and Mongolian collective ending -g. [...] A Plural -k exists also in Dravidian." '
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The plural marker -k of the Lapp substantives as an equivalent to the Finnic -t is inexplicable, cf. e.g. guolek ~ kalat "fishes" (see ibid. 208209). The supposition of the change -t > -k in this case is a phonetic-historical nonsense. The 2Psg suffix -k of the Lapp conjugation belongs here, too, thus the derivation of -k from the common with Finnic form -t (see ibid. 271) is absurd. No faith can be inspired by the vast, often multi-staged system of the derivation of the Lapp phonetic peculiarities from the Finnic-Lapp proto-language as if this were the right way to explain them. It is evident morphologically that in the regular case paradigm of substantives of the Lapp absolute declension there is no equivalent to the Finnic partitive suffix -tA (see ibid. 214216): it may be a simplification of the declension system.
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The reduction of consonants to a laryngeal stop (common feature 3 [common to Baltic Finnic and Samoyed]) is not generally typical of Finnic languages, occurring only in Finnish (e.g. tulet_tänne´ "come here!" < tule´ tänne´ < *tulek tännek) and in Estonian South-Estonian dialect (e.g. tulõ? "come!" < *tulek). Consequently it may be a relatively recent special development in Finnic languages.
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In the case of the reduction of consonants to laryngeal stops (common feature 3) one should recall Tibor Mikola's paper (1996) in which the author summarises his research results on the reduction of consonants in Uralic languages. Mikola shows that the consonants in Lapp, Mordvin and Khanty languages may have reduced to voiceless consonants or to an h-type sound, e.g. Lapp Karesuondo Vx3Pl KulléH ("to hear"). (Mikola has connected this reduction or neutralisation in Lapp with the supposition about the origin of the Lapp plural marker -k from the form -t.) I limit myself only to the evidence of the transition to an actual laryngeal stop (´) in Uralic languages, although, the consideration of Mikola's data would not substantially change the picture because, in addition to Finnic and Samoyed languages, there are again only Lapp, Mordvin and Ob-Ugric languages to be taken into account.'
So, this suffix
1) is *-k,
2) indicates plurality
3) in some languages reduces to a laryngeal *-x
4) might be substratal
Slavic has a *-ák suffix
PIE has a *-ax suffix (sometimes written *-eh2)
The PIE *-ax suffix later got its own declension (*-am etc), but was originally without one (I think; it never got one in the n.pl.)
I venture the guess that all three suffixes are one and the same, from the same language (substrate? but cf the popularity of the English diminutive suffix -y/-ie in German after the war).
Torsten