By Khan T.A., Lawson M.V.

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**Extra info for [Article] A Characterisation of a Class of Semigroups with Locally Commuting Idempotents**

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And x. are switched, l J is independent is independent the x's. = - - , l Yi that ~=i. Ykt for y = y l + Y 2 + . . +nk=n of non-negative integers and all permutations o6S k of the x's. Note that since the terms in the determinant they are alternating, coefficient so in any term where n =n , with i~j, l ] the is zero. +nk:n, those expressions by adding involve sgn(o), ni~n j for i~j. l in decreasing order, in the extra summation over all permutations The expression over of the y's. +nk=n, and all o, TCS k.

An integers) ' (all n, all s e q u e n c e s f o r m an a d d i t i v e b a s i s of r l , . t)l " n=0 i=l of A is c a l l e d i s o b a r i c 1 homogeneous weight k t (a I ) = ~ f ( a l , a 2 , . , a n ) = F ( ~ l , ~2''') of w e i < ~ t k in t h e a ' s -- the of all m o n o m i a l s ~ ir. = n. an with is t h e n u m b e r of p a r t i t i o n s rl2r 2 of the n u m b e r n. Indeed, to e a c h p a r t i t i o n rI w e can a s s o c i a t e be denoted a . n of n. an This monomial will An is a free a b e l i a n g r o u p w i t h b a s i s ~ a partition An rn the 29 At this p o i n t notation it is c o n v e n i e n t on p a r t i t i o n s .

Functions, which give in an Thus, a l - r i n g R, ~ b e c o m e s investigate s e v e r a l t y p e s of s y m m e t r i c s function. expressible for e a c h r£R, s e c t i o n we w i l l sum f u n c t i o n s h Given an e l e m e n t ~ might be described, a. is the i th e l e m e n t a r y 1 ~ is u n i q u e l y on R b y taking, In this ~I,~2 ..... ). operation and ~ b e sum h for V6R. spaces gives Just a fixed functions give for V6R, n over we w r o t e the n - f o l d a rise, n field F as p e r design, (V)= Anv.

### [Article] A Characterisation of a Class of Semigroups with Locally Commuting Idempotents by Khan T.A., Lawson M.V.

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