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2. Lexicographical order A relation ≤ on a set S is a partial order if (A1) a ≤ a for all a ∈ S (A2) If a,b,c ∈ S and a ≤ b and b ≤ c, then a ≤ c (A3) If a,b ∈ S and a ≤ b and b ≤ a, then a = b. The relation is called a total order if it satisfies the additional condition: (A4) If a,b ∈ S, then a ≤ b or b ≤ a. We write ainteger n, Nn Zn we let 0 and denote the sets of n-tuples of nonnegative integers and integers, respectively. These are abelian semigroups with respect to the operation + of ordinary vector addition. Zn Zn We define the lexicographic order on as follows. Let (i1,...,in) ∈ 0 and Zn (j1,...,jn) ∈ 0 . We define (i1,...,in) ≤ (j1,...,jn) if either (i1,...,in) = (j1,...,jn) or if (i1,...,in) 6= (j1,...,jn) and ik < jk, where k is the smallest n positive integer such that ik 6= jk. Lexicographic order is a total order on Z and, Nn by restriction, on 0 . Nn A total order ≤ on 0 is called a monomial order if it is a well-order, that is, Nn Nn every nonempty subset of 0 contains a smallest element, and if, for all a,b,c ∈ 0 with a ≤ b, we have a + c ≤ b + c. The lexicographical order is a monomial order Nn on 0 (Cox, Little, O’Shea [3, Section 2.2]). The object of this note is to show that, starting with the trivial semigroup and Nn iterating the process of adjoining an identity, we obtain the set 0 with the lexico- graphical ordering. If we adjoin infinitely many identities and infinitely many zeros to the trivial semigroup, we obtain the set Z with the usual ordering. Replacing each element of Z with a copy of Z generates Z2 with the lexicographical order- ing. Iterating this process finitely many times produces Zn with the lexicographical ordering. ADJOINING IDENTITIES AND ZEROS TO SEMIGROUPS 3

3. Iteration of the process of adjoining an identity

Let S0 = {s0} be the trivial semigroup. For s1 6= s0, let S1 = I(S0,s1)= {s0,s1} be the semigroup obtained by adjoining the identity s1 to S0. Then s0 ∗ s1 = s1 ∗ s0 = s0 and s1 ∗ s1 = s1. For s2 6= s0,s1, let S2 = I(S1,s2)= {s0,s1,s2} be the semigroup obtained by adjoining the identity s2 to S1. Then si ∗ s2 = s2 ∗ si = si for i =0, 1, 2. Continuing inductively, we obtain an increasing sequence of abelian semigroups S0 ⊆ S1 ⊆ S2 ⊆···⊆ Sk ⊆ · · · such that Sk = {s0,s1,s2,...,sk} for all k ∈ N0. Then ∞ (1) T = Sk = {s0,s1,s2,...,si,...} k [=0 is a abelian semigroup with the binary operation

si ∗ sj = smin(i,j) (1) for all i, j ∈ N0. Note that s0 is a zero in the semigroup T , but that this semigroup does not contain an identity. Define s0,i = si for all i ∈ N0 and let (0) U = {s0,0,s0,1,s0,2,...,s0,i,...}. (0) Choose s1,0 ∈/ U and consider the semigroup (0) (0) U0 = I(U ,s1,0)= {s0,0,s0,1,s0,2,...,s0,i,...,s1,0}.

For all i, j ∈ N0 we have s0,i ∗ s0,j = s0,min(i,j) and s0,i ∗ s1,0 = s0,i. (0) Choose s1,1 ∈/ U0 and let (0) (0) U1 = I(U0 ,s1,1)= {s0,0,s0,1,s0,2,...,s0,i,...,s1,0,s1,1}. Iterating this process, we obtain an increasing sequence of abelian semigroups (0) (0) (0) (0) U0 ⊆ U1 ⊆ U2 ⊆···⊆ Uk ⊆ · · · with (0) Uk = {s0,0,s0,1,s0,2,...,s0,i,...,s1,0,s1,1,s1,2,...,s1,k}. Then ∞ (1) (0) U = Uk = {s0,0,s0,1,s0,2,...,s0,i,...,s1,0,s1,1,s1,2,...,s1,i,...} k [=0 is a abelian semigroup with the multiplication

s0,i ∗ s0,j = s0,min(i,j)

s1,i ∗ s1,j = s1,min(i,j) and s0,i ∗ s1,j = s0,i 4 MELVYN B. NATHANSON for all i, j ∈ N0. Iterating this process, we obtain an increasing sequence of abelian semigroups U (0) ⊆ U (1) ⊆ U (2) ⊆···⊆ U (ℓ) ⊆ · · · with ℓ (ℓ) ∞ U = {si1,i2 }i2=0 i =0 1[ such that ∞ (2) (ℓ) ∞ T = U = {si1,i2 }i1,i2=0 ℓ [=0 is an abelian semigroup whose multiplication satisfies

si1,min(i2,j2) if i1 = j1 si1,i2 ∗ sj1,j2 = (si1,i2 if i1 < j1. Again, iterating the process of adjoining identities to semigroups, we obtain, for every positive integer n, the abelian semigroup (n) ∞ T = {si1,i2,...,in }i1,...,in=0 with multiplication ∗ defined by

si1,i2,...,in ∗ sj1,j2,...,jn = si1,i2,...,in if either (i1,...,in) = (j1,...,jn) or if (i1,...,in) 6= (j1,...,jn) and ik < jk, where k is the smallest positive integer such that ik 6= jk. The binary operation on T (n) satisfies properties (B1), (B2), and (B3), and so induces a total order on this semigroup. Since

si1,i2,...,in ∗ sj1,j2,...,jn = si1,i2,...,in if and only if

(i1,i2,...,in) ≤ (j1, j2,...,jn) with respect to the lexicographical order, it follows that the process of iterated Nn adjunction of an identity to the trivial semigroup has recreated the semigroup 0 with the lexicographical order.

4. Iteration of the process of adjoining a zero (1) We return to the semigroup T = {s0,s1,s2,..., } with the binary operation si ∗ sj = smin(i,j). Choose an element s−1 such that s−1 6= si for all i ∈ N0 and let (1) (1) (1) T1 = Z(T ,s−1) be the semigroup obtained by adjoining the zero s−1 to T . Then si ∗ s−1 = s−1 ∗ si = s−1 for all integers i ≥−1. Choose an element s−2 such N (1) (1) that s−2 6= si for all i ∈ 0 ∪ {−1} and let T2 = Z(T1 ,s−2) be the semigroup (1) obtained by adjoining the zero s−2 to T1 . Then si ∗ s−2 = s−2 ∗ si = s−2 for all integers i ≥ −2. Continuing inductively, we obtain an increasing sequence of abelian semigroups

(1) (1) (1) (1) T ⊆ T1 ⊆ T2 ⊆···⊆ Tk ⊆ · · · such that (1) Tk = {s−k,s−k+1,...,s−1,s0,s1,s2,...,si,...} ADJOINING IDENTITIES AND ZEROS TO SEMIGROUPS 5 for all k ∈ N0. Then ∞ (1) (1) ∞ V = Tk = {si}i=−∞ k [=0 is a abelian semigroup with the binary operation

si ∗ sj = smin(i,j) for all i, j ∈ Z. This operation satisfies properties (B1)-(B3) and so induces a total order on V (1). The semigroup V (1) contains neither an identity nor a zero. Z Let si1,i2 = si2 for all i1,i2 ∈ and consider the set X = {s }∞ si1 i1,i2 i2=−∞ with the total order defined by si ,i ≤ si ,j if i2 ≤ j2. Then {Xsi } ∈ (1) is a 1 2 1 2 1 si1 V family of pairwise disjoint nonempty totally ordered sets. Replacing each element s ∈ V (1) with the set X , we obtain the totally ordered set i1 si1

V (2) = X = {s }∞ si1 i1,i2 i1,i2=−∞ s ∈V (1) i1[ (2) where si1,i2 ≤ sj1,j2 in V if either i1 < j1 or i1 = j1 and i2 ≤ j2. Z Defining si1,i2,i3 = si3 for all i1,i2,i3 ∈ and replacing each element si1,i2 of V (2) with the set X = {s }∞ , we obtain the totally ordered set si1,i2 i1,i2,i3 i3=−∞

V (3) = X = {s }∞ si1,i2 i1,i2,i3 i1,i2,i3=−∞ s ∈V (2) i1,i[2 with the lexicograhical ordering. Iterating this process, we obtain, for every positive integer n, the abelian semigroup

(n) ∞ V = {si1,i2,...,in }i1,...,in=−∞ with multiplication ∗ defined by

si1,i2,...,in ∗ sj1,j2,...,jn = si1,i2,...,in if either (i1,...,in) = (j1,...,jn) or if (i1,...,in) 6= (j1,...,jn) and ik < jk, where k is the smallest positive integer such that ik 6= jk. The binary operation on V (n) satisfies properties (B1), (B2), and (B3), and so induces a total order on this semigroup. Since

si1,i2,...,in ∗ sj1,j2,...,jn = si1,i2,...,in if and only if

(i1,i2,...,in) ≤ (j1, j2,...,jn) with respect to the lexicographical order, it follows that the processes of iterated adjunction of identities and zeros to the trivial semigroup and iterated replacement of partially ordered sets produces the semigroup Zn with the lexicographical order. In this way we have created something from nothing. Remarks. For more information about the general theory of semigroups, there are excellent books by Clifford and Preston [1, 2], Howie [4], and Ljapin [5]. I wish to thank Richard Bumby for helpful discussions on this topic. 6 MELVYN B. NATHANSON

References [1] A. H. Clifford and G. B. Preston, The algebraic theory of semigroups. Vol. I, Mathematical Surveys, No. 7, American Mathematical Society, Providence, R.I., 1961. MR MR0132791 (24 #A2627) [2] , The algebraic theory of semigroups. Vol. II, Mathematical Surveys, No. 7, American Mathematical Society, Providence, R.I., 1967. MR MR0218472 (36 #1558) [3] David Cox, John Little, and Donal O’Shea, Ideals, varieties, and algorithms, second ed., Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1997, An introduction to computational algebraic geometry and commutative algebra. MR MR1417938 (97h:13024) [4] John M. Howie, Fundamentals of semigroup theory, London Mathematical Society Mono- graphs. New Series, vol. 12, The Clarendon Press Oxford University Press, New York, 1995, , Oxford Science Publications. MR MR1455373 (98e:20059) [5] E. S. Ljapin, Semigroups, third ed., American Mathematical Society, Providence, R.I., 1974, Translated from the 1960 Russian original by A. A. Brown, J. M. Danskin, D. Foley, S. H. Gould, E. Hewitt, S. A. Walker and J. A. Zilber, Translations of Mathematical Monographs, Vol. 3. MR MR0352302 (50 #4789)

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