Showing papers on "Idempotence published in 1979"

Fully idempotent factorization rings

Abstract: (1979). Fully idempotent factorization rings. Communications in Algebra: Vol. 7, No. 6, pp. 547-563.

Compelled operations and operations of degreeP

Abstract: We definen-ary (n ≥ 1) operations compelled by an operator and operations of degreep, generalizations of Greibach's binary syntactic operations. We show that properties of binary syntactic operators remain right forn-ary operations (n ≥ 1) and further for idempotent operations. We obtain so hierarchy theorems among full semiAFLs and new properties about classical operations and classical families of languages. The new definitions and properties are illustrated by study of a family ofn-ary operations.

On transitive commutative idempotent quasigroups

Abstract: Commutative idempotent quasigroups with a sharply transitive automorphism group G are described in terms of so-called Room maps of G. Orthogonal Room maps and skew Room maps are used to construct Room squares and skew Room squares. Very general direct and recursive constructions for skew Room maps lead to the existence of skew Room maps of groups of order prime to 30. Also some nonexistence results are proved. 1980 Mathematics subject classification (Amer. Math. Soc.): primary 05 B 15; secondary 20 N 05. 1. Room squares and orthogonal ci-quasigroups 1.1. Let r be an odd integer. A Room square of side r is an arrangement of r+1 distinct objects in a square array of side r satisfying (i) each of the r cells of the array is either empty or contains exactly two distinct objects, (ii) each row and each column of the array contains each of the r+\\ objects exactly once, (iii) every unordered pair of distinct objects occurs in exactly one cell of the array. The square is skew, if, in addition, (iva) cell (/, 0 contains the pair i, oo, (ivb) cell (i,k) is empty if and only if i^k and cell (k,i) is not empty; here cell (/, k) is the cell in row / and column k. A Room square is known to exist if and only if r^3,5; see for example Wallis (1973a, 1974). 411 use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S1446788700013422 Downloaded from https://www.cambridge.org/core. IP address: 54.70.40.11, on 17 Apr 2019 at 18:46:58, subject to the Cambridge Core terms of 412 Arnold Neumaier [2] 1.2. We refer to a commutative idempotent quasigroup (Q, *) as a ci-quasigroup. Two ci-quasigroups (Q, *) and (Q, **) are orthogonal if and only if the equations x*y = a, x**y = b have at most one solution {x,j}£Q (as unordered pair), for every a,beQ; they are skew orthogonal if and only if, in addition, x = y = z = t is the only possibility to satisfy the equations x*y = z**t, x**y = z*t. Note that the two quasigroups are never orthogonal if considered simply as quasigroups. 1.3. By row and column permutation and renumbering of elements we may standardize any Room square such that the diagonal cells (i,i) contain the pair /,oo, where oo is a fixed element. If we define two operations * and ** on the set g = {l,...,/•} by x *y = z if and only if x = y = z, or x^y and the pair x,y is in row z, x **y = z if and only if x = y = z, or x^y and the pair x,y is in column z, a simple verification shows that (Q, *) and (Q, **) are a pair of (skew) orthogonal ci-quasigroups if and only if the given square is a standardized (skew) Room square. Conversely, from (skew) orthogonal ci-quasigroups (Q, *), (Q, **) one may construct a standardized (skew) Room square, defining x, oo is in cell (x, x) for every xeQ, x,y is in cell (x*y,x**y) for every pair x,ysQ,x^y, the other cells are empty. The proofs are straightforward and thus omitted; see Brack (1963). 2. Room maps and sharply transitive ci-quasigroups A ci-quasigroup (Q, *) is sharply transitive if and only if it possesses a group G of automorphisms, sharply transitive on Q. We describe sharply transitive ciquasigroups within the group G. 2.1. Let G be a finite group. We call a map G a Room map if and only if 0) P ( l ) = l . (2) ?(x) = cp(x) (xeG), (3) { = l (x,yeG) then

)=>y = x or j ; = x' (x,yeG) then we say * = .>'=l (x,yeG) then 9?x and f>2 are said to be •s&evv orthogonal. From the definitions we see immediately that (p is strong if and only if

= xv (equivalently, u = xw and v = ^ ^ w ) . Since G has odd order, the map x->x is a permutation of G, thus *p is well defined. Because of