▁to ▁clarify ▁and ▁ref ine ▁White head ' s ▁theories . ▁The ▁domain ▁for ▁both ▁theories ▁consists ▁of ▁" reg ions ." ▁All ▁un quant ified ▁variables ▁in ▁this ▁entry ▁should ▁be ▁taken ▁as ▁t ac it ly ▁univers ally ▁quant ified ; ▁hence ▁all ▁ax i oms ▁should ▁be ▁taken ▁as ▁universal ▁clos ures . ▁No ▁ax iom ▁requires ▁more ▁than ▁three ▁quant ified ▁variables ; ▁hence ▁a ▁translation ▁of ▁first ▁order ▁theories ▁into ▁relation ▁algebra ▁is ▁possible . ▁Each ▁set ▁of ▁ax i oms ▁has ▁but ▁four ▁exist ential ▁quant ifiers . ▁ ▁In clusion - based ▁point - free ▁geometry ▁( m ere ology ) ▁The ▁ax i oms ▁G 1 - G 7 ▁are , ▁but ▁for ▁number ing , ▁those ▁of ▁Def . ▁ 2 . 1 ▁in ▁Ger la ▁and ▁Mir anda ▁( 2 0 0 8 ) ▁( see ▁also ▁Ger la ▁( 1 9 9 5 )). ▁The ▁ident ifiers ▁of ▁the ▁form ▁W P n , ▁included ▁in ▁the ▁ver bal ▁description ▁of ▁each ▁ax iom , ▁refer ▁to ▁the ▁corresponding ▁ax iom ▁in ▁Sim ons ▁( 1 9 8 7 : ▁ 8 3 ). ▁ ▁The ▁fundamental ▁primitive ▁binary ▁relation ▁is ▁In clusion , ▁denoted ▁by ▁inf ix ▁" ≤ ". ▁( In clusion ▁corresponds ▁to ▁the ▁binary ▁Par th ood ▁relation ▁that ▁is ▁a ▁standard ▁feature ▁of ▁all ▁mere ological ▁theories .) ▁The ▁intuit ive ▁meaning ▁of ▁x ≤ y ▁is ▁" x ▁is ▁part ▁of ▁y ." ▁Assuming ▁that ▁identity