### Table 1: Semigroups satisfying x

"... In PAGE 2: ... 3. Is the congruence #18 m;n on A #03 decidable? In other words: Is the free word #28iden- tity#29 problem for the variety de#0Cned by #28associativityand#29x m+n = x n recursively solvable? The top-left case #28m =1,n =0#29of Table1 is a trivial algebra. Thenextcaseof the top row#28m; n = 1#29, the #28free#29 idempotent semigroups, are called #28free#29 bands.... In PAGE 3: ... Green and Rees #5B 1952 #5D proved that x m = 1 is #0Cnite if and only if the semigroup x m+1 = x is. So columns 1 and 2 of Table1 haveidentical #0Cniteness properties. In particular, bands #28xx = x#29 are #0Cnite.... ..."

### Table 1. Sizes of some investigated semigroups.

2003

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### Table 1: Examples of intermediate free semigroups and bases of identities for V

### Table 3. Regularity priorities.

2002

"... In PAGE 3: ... The hierarchies are truncated by detecting a large jump in the tolerance values between the clusters. The regularities with the priorities (see Section 3) are listed in Table3 . Each regularity is described by a constraint set.... In PAGE 5: ... For planar symmetrically arranged directions, we put the main empha- sis on the number of occupied positions and faces arranged in a loop as computed by wra(r) and wa(F; 1), but also consider the special angle value for the planar arrangement and the geometric types involved. Table3 lists the constants for wc and wb, and the way wq is computed for the regularity types. wq, wc and wb were derived from a part survey estimating the frequency of regularities in simple mechanical components [15].... ..."

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