Any set throughout whole this paper will be assumed non-empty, if it will be not specially given a contrary and any order type, i.e. a type of linearly ordered set, will be assumed as a type of a set containing at bast two elements. If G is a set, then card a denotes the cardinality of G . A linearly ordered set will be called a chain, a set ih which every two distinct elements are incomparable will be called an antichain.
We shall use the BIRKHOFF'S symbolik [l] so that for instance 2 denotes a type of a chain containing two elements whereas 2 denotes a type of an antichain containing two elements. We shall not distinguish between ordered sets and types of these sets. Hence for instance WE denotes both the cardinal power1) in which basis is a well-ordered set of type w , and exponent is an antichain containing two elements, i.e. the set of all pairs [x, y] where 0 5 x, y < w, , [z, y] 5 [ d , y'] e+ x I d, y(= y' and the type of this cardinal power. D e f i n i t i o n 1. Let G be an ordered set, let a be an order type. Let L be any chain of type a . Let K be a set, let f, be an isoton mapping (one-one isoton mapping) of the seb G into the set L far every x E K . If for any two elements x, y E G there holds the following statement: x y if and only if f x ( z ) 2 f,(y) for every x E K , then we shall say that the system {f, I x E K } of mappings of G into L is a-pseudorenlixer (a-realizer) of the set G. By a cardinwlity of this a-pseudorealizer (a-realizer) will be understood the cardinality cardK.
It is known ([4], Corollary2.1.) that to every ordered set G and every order type a there exists a t least one a-pseudorealizer of G . But there need not exist an a-realizer of a. A necessary, but not sufficient condition for the existence of an a-realizer of G is cardG 5 carda.2) D e f i n i t i o n 2. Let G be an ordered set, let a be an order type. By the a-pseudodimension of the set C (a-pdimG) will be understood the minimum of cardinalities of all a-pseudorealizers of the set G . If there exists a t least one u-realizer of the set G , then we define the a-dimension of G (a-dimG) as the mimmum of cardinalities of all a-realizers of the set G .
l) The definition of cardinal'power 'can be found in [2].
2, Let A be the order type of the set of all real numbers. Let B be the ordinal power i.e. the set of all pairs [x, y] , where 0 2 x, y < w1 and [z, y] < [x', y'] ++ z < z' or x = x', y < y' .
Then cardB = = N, 5 2 b = cardil. But there doesn't exist any l-realizer of G because G contains a non-denumerable system of non-overlapping intervals while any system of nonoverlapping intervals in any linearly ordered set L of type 1 is countable.