Let's have the positive first to establish the unmarked word order:
lyase ruve gonyen barat woman old.person stick-ins walk-3s-PRS
"The old woman walks with a stick"
Note that "with" is expressed by the instrumental case, meaning that the stick is used as a tool for walking. Expressed by the preposition az, which has comitative meaning ("together with"), it would seem like the old woman and the stick are walking side by side, as if they were sisters:
? lyase ruve az gonyen barat woman old.person with stick-dat walk-3s-PRS
"The old woman walks in the company of a stick"
Let's negate something; as we want to say "without a stick" we'll negate the stick.
ni gonyen lyase ruve na barat NEG stick-ins woman old.person NEG walk-3s-PRS
"It is not with a stick that the old woman walks"
This seems to imply that she walks, for instance, with the branch of a tree or the back of a chair. Perhaps a less marked word order would help:
lyase ruve ni gonyen na barat woman old.person NEG stick-ins NEG walk-3s-PRS
"The old woman doesn't walk with a stick"
It doesn't help much; the nominal negation ni tends to slightly mark whatever it applies to, and we still want to ask "then what does she walk with?" Let's try it without ni:
lyase ruve gonyen na barat woman old.person stick-ins NEG walk-3s-PRS
"The old woman doesn't walk with a stick"
The voice in the back of my head now asks "but what does she do with the stick then, beat the dog?" We know that she walks; let's make her walk but not use the stick:
lyase ruve ni gonyen barat woman old.person NEG stick-ins walk-3s-PRS
"The old woman walks, not with a stick"
Better; much better, even though it goes against the Ilaini school-grammar rule that the verb should always be negated if anything else in the sentence is. It does still imply that she uses something else to walk with, though. It seems that we need 'without' after all.
lyase ruve gonyin liz barat woman old.person stick-abl without walk-3s-PRS
"The old woman walks without a stick"
Ah, that's it. Perfectly unmarked and it means exactly what we want. It can never mean "not accompanied by a stick"; the sentence doesn't tell whether or not the woman has a stick with her, only that she isn't using one to walk. And we can't say this either:
* lyase ruve parnei liz barat woman old.person sister-abl without walk-3s-PRS
"The old woman walks without her sister"
because that would mean that the old woman uses her sister as a tool (or rather, doesn't do it, but negating something implies that the positive also exists). We have to say
lyase ruve parnen niez barat woman old.person sister-dat NEG.with walk-3s-PRS
"The old woman walks unaccompanied by her sister"
If we negate the verb in this sentence it does change the meaning (that is, niez doesn't count as a negation that requires a negated verb; it's more like the comitative equivalent of instrumental liz):
lyase ruve parnen niez na barat woman old.person sister-dat NEG.with NEG walk-3s-PRS
"The old woman doesn't walk unaccompanied by her sister"
implying that she doesn't (want to) take a walk if her sister doesn't come along.
Another use of liz, probably the original one from which the postposition developed, is as a conjunction:
Le fulut farey liz laynynat RFL take.away-PRT-3s goodbye without speak-PNC-PRS-3s
"She left without saying goodbye"
(literally: "She took herself away without she-says may-you-reach-your-goal")
The use of the present tense and imperfective aspect (with zero marker) in the subordinate clause indicates that the action takes place at the same time as that of the main clause.
Using the perfective aspect (laynynenat, where -yn- would probably be elided) would make it mean "She left without having said goodbye". This makes sense, but means something different, most probably that she was making elaborate goodbyes and had to leave suddenly before she could finish it.
The preterite in the subordinate clause would imply that she left without having said goodbye earlier (a kind of pluperfect) or having ever said goodbye on any occasion that she could have.
Note that liz, joining farey laynynat to le fulut, appears in the proper place for a clause conjunction: after the first constituent (the object phrase). Compare:
Le fulut farey so laynynat take.away-PRT-3s goodbye and speak-PNC-PRS-3s
"She left and said goodbye", or "She left, saying goodbye".
Liz as a conjunction can also mean "except when" or "unless", as in the Charter of the Guild of Anshen: dilynet liz mustyen codien "except when it happens in a fair fight".