On reconstruction of monoids from their table fragments

Abstract

where ix, x j, xij~X. The elements xx . . . . . x, will be called the elements in the outer row, ix,. . . . . x in the outer column and all the remaining elements the inner elements of the table. Since the inner elements form a n • n-matrix we shall speak about its rows and columns as about the inner rows and the inner columns. Two tables will be called equivalent if they belong to the same monoid, two tables will be called weakly equivalent if they belong to the isomorphic monoids. Omitting some elements in the multiplication table we get a partial table of a monoid. Our main concern will be with the question, how many elements we can omitt in the table under the condition that the monoid could be uniquely reconstructed from the partial table. Having a set X we can construct a partial table over X. By it we mean a rectangular scheme of the type (1) when some of the elements may be omitted. We see three questions connected with partial tables over X.