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On right residuals in lattice ordered groupoids

✍ Scribed by Ferenc A. Szász

Publisher
John Wiley and Sons
Year
1972
Tongue
English
Weight
381 KB
Volume
53
Category
Article
ISSN
0025-584X

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✦ Synopsis

By FERENC A. Sziisz of Budapest To Professor LBSZL~ KALW~R on his 65t" birthday (Eingegangen am I S . 3. 1971)

Following G. BIRKHOFF [2] and I,. FUCHS [3, p. 1911 a lattice ordered groupoid, or shortly a 1. 0. groupoid, is defined as a groupoid G, which is a t the same time also a lattice, satisfying the distributivity requirements : ('1 ( u w b ) c = u c w b c and u ( b w c ) = u b w u c for every u, b, c E G. 'Ihe nionotoiiity laws, i. e. u 5 b always implies LL c 5 b c and c u 5 c b, follow obviously from (1). Here the duals of (1) &re neither assumed, nor can be derived from (I), therefore for 1. 0. groupoids the duality principle fails generally to be valid. If we assume that for all ci, b E G an element c = u : b there exists such that, (2) x Ic is equivalent to b x 5 a , then c = u : b is called the right residual of cc and b, furthermore G! is said to be a right residuated 1. 0. groupoid. More generally, u : b can be defined also in some partially ordered groupoids. The right residual generalizes the cboncept of the right-sided ideal quotient in the ring theory. Some properties of 1.0. groupoids and of right residuals are collected in G. BIRKHOFF [2] a i d L. FUCHS [3].

If the 1. 0. groupoid 6: is a complete lattice and it fulfils the infinite distributive lams :

then G is called a complete 1. 0. groupoid. For results on some 1. 0. groupoids me refer the reader e. g. to B. A. L. LESIEUR [7], 0. STEINFELD [S, 9, 101, I. V. STELLECKIY [Ill, E. G. SUL'-GEYPER [12], F. s z h z [13], M. WARD [I71 andM. WARD-R. P. DILWORTH [IS]. For notions of lattice theory see yet G. S z h z [16]; furthermore for notions of ring theory and group theory see N. JACOBSON [4] and A. G. KUROSH [ 5 ] ,

respectively.

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