The kind of tack to use has nothing to do with the breed of the horse. Conformation, size, level of training, temperment, the horse's work or sport are the defining factors in what tack you will use.
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dressage
This is a type of tack that is used on the horses eyes to limit eyesight.
Usually english. This breed of horse excels at dressage.
By equipment I assume you mean tack. The halter is used to lead, lounge or tie your horse. The bridle is a headstall, bit and reins and this helps the rider direct the horse where he wants the horse to go. The saddle, whether western or english, is strapped to the horse's back and where the rider sits.
Martingale
Many tack stores will also have used equipment on consignment. Check your local tack stores and ask about any items on consignment. It's a great way to aquire usable tack at discount prices.
A halter is the basic piece of tack used to move your horse from one place to another.
The saddlery used on a horse is called tack, and this may include saddles, bridles, saddle pad, halter, etc.
A horse can throw it's head back no matter what piece of tack is used if it's determined to do it. But many people use a martingale to keep a horses head down, though this is not a good substitute for real training of the horse.
When You Walk a horse out on the streets, you must have a bridle on the horse in order to controll it. that is all you need but you can fully tack up if yu wish...just to get your horse used to the saddle.
In most areas you can find used or consignment tack shops where you can get great deals on 'preowned' horse tack. There are also numerous places online where you can shop and compare from your own home.
It's called a bearing rein but it causes severe pain to the horse and I strongly recommend you don't use it.