Introduction
In the present paper a multiplicative system, called a pseudocategory, is introduced. This notion is very useful in the algebraic theory of perturbations of Linear operators. The first attempt in this direction was given in the monograph [ 13 of D. PRZEWORSKA-ROLEWICZ and S. ROLEWICZ, where there were defined "pararings" and "paraalgebras" for operators mapping a linear space into another. A natural generalization of these notions is the notion of a pseudocategory. An axiomatic of pseudocategories was given by D. PRZEWORSKA-ROLEWICZ in [l]. In her paper [2], in a rather complicated way, properties of pseudocategories have been studied. It was shown that pseudocategories can be extended to concrete categories under some assumptions. Furthermore pararings and paraalgebras, ideals and radicals were defined by means of pseudocategories. Then, in a similar way, as in the book mentioned, the algebraic theory of perturbations of linear operators over a class of linear spaces were examined.
It should be pointed out that in the theory of pseudocategories the notion of objects does not play any role. The axiomatic contains only the notion of morphsms. The existence of a unit is also not necessary. This is natural, in a sense. For instance, in a quotient paraalgebra of linear operators over a class of linear spaces, the spaces under consideration are not objects. One can introduce objects in this case, but in an artificial way.
The theory of pseudocategories with the axiomatic given by the second of the present author in [I], [2] has been simplified in an essential way by H. LAUSCH in 1985. The present paper contains the theory of pseudocategories and pararings modified by the first author and the perturbation theory of linear operators over a class of Linear spaces and some results concerning paraalgebras induced by a right invertible operator. Contents Math. Nachr. 138 (1988) 3. Paraalgebras and perturbations of linear operators 4. D-parccalgebras References 1.
Pseudocategories
The following definition is fundamental for all subsequent considerations : Definition 1.1. A class P of morphisms is called a pseudocutegory if for some ordered pairs (x, y ) of morphisms x, y E P a pTOd7LCt z E P is defined, denoted by z = zy, such that for all a , b, c, d E P Al: if ab, cd, cb exist then ad exists; A, : if ab, bc exist then (ab) c, a(bc) exist and (ab) c = a(bc) ; cul2 it abc. A,: if ab, (ab) c exist then bc exist; if bc, a(bc) exist then ab exists; A,: (z E P : ax, xb exist} i s a set. called a multiplicant and is denoted by k i a l b . Proposition 1.1. Let P be a pseudocategoy. Definition 1.2. If P is a pseudocategory and a, b E P then (x E P : ax, xb exist] is (i) If a, 6, b,6, x, y E P and ax, iix, yb, y6 exist then MaIb = Malb -* 9 (iii) if c, u, v E P and uv exists then M,,, = M,I,, and Mu,, = Mu,,,; (iv) if a, x, y , z E P and x, y E Mala then the existence of zx implies the existence of zy Proof. (i) Let z E Map. Then zb, y8, yb exist. Hence, by A,, the product & exists. Also Zx, az, a x exist. Hence, by Al, the product 6z exists. Therefore, z E Mzlij, which implies ikt(,lb E Mzli. By symmetry, Malb = Mzlg.