Hi Roland, thanks for the question.

Arpeggios work the same in all situations, and with all DAWs, so nothing different for Logic, from Cubase, or from Sonar, or Any other DAW.
In order for an Arpeggio to begin requires the following situation: the Main Arp On/Off switch must be On, the Part Arp On/Off switch must be armed, and a note (or notes) must trigger the keys zoned to control the arp at the appropriate velocity. The notes can originate locally from the MOXF key bed or via MIDI In. Once those conditions are met the arp phrase will begin. The Arpeggiator can be set LOOP or play through the assigned phrase just once.

In order for an Arpeggio to end (stop) you must meet the requirements set forth in the arp's setup...
_ If the arp is set to HOLD = Off, or HOLD = Sync-Off, the arp will stop as soon as the note-on events set to trigger it are removed.
_ If the Arp is set TRIGGER Mode = Gate and it is set to HOLD = On it will continue to play until you give it a command to stop.
_ If the TRIGGER mode = Toggle, another note will toggle the Hold Off (the note must be enter via the keyboard, not through Midi)

Although you don't mention the settings you have currently, if your arp is not stopping when you stop your sequencer (Logic), we can easily conclude a few things:
1) You are triggering the arp via MIDI from note-on events from the DAW. The arpeggiators are armed.
2) HOLD is On, TRIGGER mode is Gate

Is this what you want to do?

Before we make recommendations to create a stop message, we have to ask, why you did not record the ARPEGGIO OUT to your Midi Track? Doing so transfers the arpeggio phrase data to actual MIDI Events on your track same as if input via the keyboard. Recording the arp data means it becomes like all other note-on events recorded to tracks of the DAW. You can disarm the Arpeggiators on playback and never have to worry about the Arpeggiator continuing endlessly.

This is not a rhetorical question, because you may have good reason, but we need to mention it because, it is the provided solution: to record the arp phrase as standard MIDI events to a track, this way, it frees the arp for new duties, it allows you to edit the individual note event data, and it eliminates issues having to do with stopping it at the end of the composition.

Once the arp is recorded, you no longer need the Arpeggiator to be armed.

If you do require the Arps to be active/live, or simply because setting up to record Arps can be tricky; (recording arpeggios, as you may have guessed does require a different setup from "normal" recording); you can Stop the arpeggio from an external sequencer by controlling it with your MIDI messages:

_ depending on what you are doing, you could set HOLD = Off, and manually lift your hand at the point you want the arp to stop playing. The actual duration (gate time) that you hold the controlling/triggering keys will be documented. When ARP HOLD is On, it takes the place of your hands holding down the keys. So unless you document the lifting of your hands, the tone generator still "sees" or "thinks" you are holding the Keys down.

It does respond directly to messages from the keys. And unlike other controllers, that are forced to reset when the Stop command is sent, the arpeggiators work as a live function in response to the keys... So it physically works.

_ if that is not an attractive option (because perhaps the time of the use of the Arpeggiator is too lengthy to hold with your hands, you can "hold" the arp using the sustain pedal. Or simply sustain pedal messages. Set Arp HOLD = Off, and use the sustain pedal to hold the triggering notes. Remember, Arp Hold takes the place of you physically having to hold the keys, this is exactly what the Sustain pedal does in MIDI. So you could, with this Sustain messages stop the arpeggio by inserting a simple cc064 = 0 Sustain pedal Off message at the appropriate measure and beat. Also with the Arp HOLD set Off you don't have to worry about it running forever. You control it with precise commands. Treat it like you would piano sustain.

or if you simply want to automate it... You could use a custom System Exclusive message to turn the Part Arp Switch to Off at the end of the composition
F0 43 10 7F 1C 00 38 pp 00 dd F7

pp is the Part number 00-0F
dd is the Data Byte 00 is Off, 01 would be On

So to turn the Part Arp Switch = Off on Part 1 the message would be:
F0 43 10 7F 1C 00 38 00 00 00 F7

WOW! thank you so much "Good Mister" 🙂 for all this informations! i will check them out and reply asap. have to prepare some stuff for a gig, but tomorrow i will have time to set up everything in a good way.

SOLVED - 🙂 to make it quick: the walktrough:
(ARP is ON)

1.) LOGIC PRO X (LPX):
SETTINGS - MIDI - SYNC - MIDI-Sync Settings (new window opens:)
synchronisation: MIDI - MIDI-CLOCK send to "Ziel 1" Yamaha MOXF6 Port 1"

2.) MOXF:
UTILITY - SYNC:
MIDI Sync: auto OR external (see p. 148 ref.manual) external: a must, if the arp should be controlled EXKLUSIV by seq (LPX) (not by moxf)
ClockOut: off (!) comment: tempo of arpeggio will be defined by bpm setting in logic (MOXF = Slave)
SeqCtrl: in (!) comment: arpeggio will start and stop (!) related to LPX start stop command (see ref manual page 149) - no sysex needed

3.) (used performance: Film Noir)
F3 ARP ED:
MAIN: Hold: OFF (!)

F5 PLAY FX: go to next page (-> ->):
LOOP: ON
TriggerMode: gate

as mentioned by BadMister:
- push a key to start the sequenze (arp)
- hold sustain pedal to keep it go
- lift the sustain pedal to make it stop.

RECORDING INTO LOGIC:
- lifting sustain pedal will not be necessary if the arpeggio has to flow the whole song 🙂 as a sequencer-stop also stops the arpeggiator.

SO! this should solve my problem - please check it out yourself - perhaps there are some correction/enhancements to make 😉