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Tensor Products of Ternary Semimodules over Ternary Semifields

Written By

Thorranin Pawaputanon Na Mahasarakham, Ruangvarin Intarawong Sararnrakskul and Nissara Sirasuntorn

Reviewed: 30 November 2022 Published: 29 December 2022

DOI: 10.5772/intechopen.109242

Optimization Algorithms - Classics and Last Advances IntechOpen
Optimization Algorithms - Classics and Last Advances Edited by Mykhaylo Andriychuk

From the Edited Volume

Optimization Algorithms - Classics and Last Advances [Working Title]

Dr. Mykhaylo I. Andriychuk and Dr. Ali Sadollah

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Abstract

We introduce tensor products of ternary semimodules over ternary semifields and the n-fold tensor products for general case. We prove the universal mapping property of the n-fold tensor products. Moreover, we later provide a condition for preserving the flatness of ternary semimodules with tensor products of ternary semimodule homomorphisms.

Keywords

  • tensor product
  • ternary semimodule
  • ternary semifield
  • ternary algebra
  • abelian groups

1. Introduction

The subject of ternary algebra was started by D. H. Lehmer [1]. He studied some concepts of ternary systems called triplexes, which obtained the generalization of abelian groups. Afterward, Los [2] mentioned this algebraic structure studied by Banach and showed an example of a ternary semigroup, which is not a semigroup. In 1971, W. G. Lister [3] studied the abstract structure of a ternary ring, which is a ternary product of abelian groups. In 2003, R. Intarawong [4] studied and provided the universal mapping property of tensor products of modules over semifields. T. K. Dutta and S. Kar [5] studied ternary semiring and gave some properties of ternary semifields. In addition, H. J. M. Al-Thani [6] studied the flat semimodules, which is constructed by the exact sequences of semimodules and tensor products of semimodule homomorphisms.

The universal mapping property in the branch of modules over rings gives that there exists a unique group homomorphism from MRN to a module A, which is composed with a bilinear map M×NMRN is a bilinear map M×NA. The structure of the ternary semimodules also leads us to derive a similar result.

The goal of this research is to investigate some properties of ternary semimodules over ternary semifields. We also study the universal mapping property of tensor products of ternary semimodules and investigate some types of ternary semimodules defined by tensor products of homomorphisms on ternary semimodules and exact sequences.

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2. Preliminaries

The following familiar definitions and theorems from Refs. [7, 8, 9] regarding the notion of free abelian groups are needed to define tensor products of ternary semimodules over ternary semifields.

A ternary semifield K is a system K+ with a binary operation (+) and a ternary operation satisfying the following conditions for all u,v,w,s,tK.

  1. K+ is a commutative semigroup with identity 0,

  2. uvwst=uvwst=uvwst,

  3. u+vws=uws+vws,

  4. uv+ws=uvs+uws, and.

  5. uvw+s=uvw+avs.

  6. uvw=vuw=wvu=uwv,

  7. 0Ku,vK,0+u=u and 0uv=u0v=uv0=0, and.

  8. uK\0vKtK,uvt=vut=tuv=tvu=t.

(An element v is called an inverse of u. In addition, the inverse of u is unique and u1 denotes the inverse of u.)

For convenience, let uvw denote uvw for all u,v,wK.

Let R+ be a commutative ring. For a nonempty set S of R, a subring S of R is said to be a positive cone if SS=R and SS=0. S is called a negative cone of S.

It is clear that every negative cone of a cone S of R is a ternary semiring. For example, Z0 is a natural example of a ternary semiring. Moreover, Q0 and R0 become ternary semifields.

A group F is a free abelian group if F is an abelian group and for every nonzero element g of F, there exist unique nonzero integers α1,α2,,αn and unique distinct x1,x2,,xn in XF such that g=α1x1+α2x2++αnxn. We sometimes call X a basis for F.

For a nonempty set X, let

FAX=f:XZFXsuch thatF<andfx=0forallxX\F.E1

Define + on FAX by for any f,gFAX,

f+gx=fx+gxforallxX.E2

Then, FAX+ is an abelian group. For any xX, define fx:XZ by

fxy=1,ifx=y,0,otherwise.E3

Then, fxFAX for all xX. A group FAX+ is a free abelian group on fxxX (FAX). We sometimes say instead that FAX is a free abelian group on X.

Let K be a ternary semifield. A left K-ternary semimodule or left ternary semimodule over K is an additive abelian group M together with a function from K×K×M into M, defined by k1k2mk1k2m called ternary scalar multiplication, which satisfies the following conditions for all m,m1,m2M and k1,k2,k3,k4K,

  1. k1k2m1+m2=k1k2m1+k1k2m2,

  2. k1k2+k3m=k1k2m+k1k3m,

  3. k1+k2k3m=k1k3m+k2k3m,

  4. k1k2k3k4m=k1k2k3k4m=k1k2k3k4m.

A right K-ternary semimodule or right-ternary semimodule over K is defined similarly via a function M×K×K into M and satisfies the obvious analogs of (1)–(4).

For convenience, we simply write KM [MK] as M is a left [right] ternary semimodule over a ternary semifield K.

From now on, unless specified otherwise, “K-ternary semimodule KM [MK]” means “left [right] K-ternary semimodule M .” Moreover, “K-ternary semimodule” means “left K-ternary semimodule.”

Example 1.1 Let F01=ff:01Q0 with the operations + and , defined by.

f+gx=fx+gxandαβfx=αβfxforallx01,E4

where f,gF01 and α,βQ0. Then, F01 is a left Q0-ternary semimodule.

Example 1.2 If n and M1,M2,,Mn are ternary semimodules over a ternary semifield K; then, M1×M2××Mn is a ternary semimodules over K under usual componentwise addition and scalar multiplication.

Let M be a left [right] ternary semimodule over a ternary semifield K. A left [right] ternary subsemimodule of M is a subset of M, which is itself a left [right] ternary semimodule over K with the addition and ternary scalar multiplication of M.

Let K and S be ternary semifields. An additive abelian group M is a KS-ternary bisemimodule if M is a left K-ternary semimodule and also a right S-ternary semimodule, and k1k2ms1s2=k1k2ms1s2 for all k1,k2K,mM, and s1,s2S. We write KMS for a KS-ternary bisemimodule M.

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3. Universal mapping properties

For given ternary semimodules M and N, over a ternary semifield K, it is known from Example 1.2 that M×N is a ternary semimodule over K. To find another ternary semimodule over K, arising from M and N, which is different from M×N, the tensor product of M and N is the case. The notion of free abelian groups plays a major role in constructing the tensor product of ternary semimodules over K.

Let MK and KN be ternary semimodules over a ternary semifield K. For each mM and nN, a function fmn:M×NZ is defined by

fmnxy=1,ifxy=mn,0,otherwiseE5

for all xMK and yKN. Then, fmnFAM×N and FAM×N is a free abelian group on a basis fmnmMnN.

Definition 1.3 Let MK and KN be ternary semimodules over a ternary semifield K, and let F be the free abelian group on M×N, that is, F=FAM×N. Let L be the subgroup of F generated by elements of the following forms:

  1. fm+m˜nfmnfm˜n,

  2. fmn+n˜fmnfmn˜,

  3. fmαβnfmαβn

where α,βK, m,m˜M, and n,n˜N. We call F/L the tensor product of M and N, and denoted by MKN.

Note 1.4 If MK and KN are ternary semimodules over a ternary semifield K. Then,

  1. MKN+ is an abelian group.

  2. fmnF for all mMK and nKN.

Recall that F=FAM×N and F/L=f+LfF. For any mM and nN, we write mn for fmn+L. Moreover, for all f+LF/L,

f+L=mnFM×Nαmnβmnmn,E6

where αmn,βmnZ and F<, and we simply write f+L as

i=1pαmnβmnminiori=1pαiβiminiE7

where αi=αmini,βi=βminiZ, miM, and niN.

The following property can be derived directly from Definition 1.3.

Let MK and KN be ternary semimodules over a ternary semifield K. Then,

  1. m+mn=mn+mn,

  2. mn+n=mn+mn,

  3. mαβn=mαβn,

  4. m0=0n=00=0,

for all α,βK, m,mM and n,nN.

Let MK and KN be ternary semimodules over a ternary semifield K, and A is an additive abelian group. A middle linear map (over K) from M×N to A is a function τ:M×NA such that for all m,m˜M, n,n˜N, and α,βK,

  1. τm+m˜n=τmn+τm˜n,

  2. τmn+n˜=τmn+τmn˜, and

  3. τmαβn=τmαβn.

Let MK and KN be ternary semimodules over a ternary semifield K. The map μ:M×NMKN defined by μmn=mn is called the canonical middle linear map. The function π:FAM×NFAM×N/L defined by πx=x+L for all xFAM×N is an epimorphism of groups, which is called the canonical projection.

Lemma 1.5 ([9], p. 43): Let A and B be additive abelian groups. If f:AB is a group homomorphism, and C is a subgroup of kerf; then, there is a unique group homomorphism f̂:A/CB such that f̂a+C=fa for all aA, imf̂=imf and kerf̂=kerf/C. Moreover, f̂ is a group isomorphism if and only if f is a group epimorphism, and C=kerf. In particular, A/kerfimf.

By making use of Lemma 1.5, we get the following theorem.

Theorem 1.6. Let MK and KN be ternary semimodules over a ternary semifield K and μ:M×NMKN be the canonical middle linear map. For any additive abelian group U over K, and any middle linear map ψ:M×NU, there exists a unique group homomorphism ψ˜:MKNU such that ψ=ψ˜μ, that is, the following diagram commutes.

Proof: Let U+ be an abelian group. By Theorem 1.1 ([9], p. 71) there exists a unique group homomorphism ψ̂:FAM×NU such that ψ=ψ̂φ.

Let L be the subgroup defined in Definition 1.3. Then, L is a subgroup of kerψ̂ because ψ is a middle linear map and ψ=ψ̂φ.

Let π:FAM×NFAM×N/L be the canonical projection. Since L is a subgroup of kerψ̂, by Lemma 1.5, there exists a unique group homomorphism ψ̂:FAM×N/LU such that ψ̂=ψ˜π. Now, we obtain a group homomorphism ψ˜:MKNU. Next, we will consider the diagram.

For each mM and nN, the canonical middle linear map μ:M×NMKN satisfies μmn=mn=φmn+L=πφmn=πφmn. That is, μ=πφ. Hence, ψ˜μ=ψ˜πφ=ψ˜πφ=ψ̂φ=ψ. Last, let ρ:MKNU be a group homomorphism such that ψ=ρμ, and let θ=ρπ. Consider the following diagram.

We have that θφ=ρπφ=ρπφ=ρμ=ψ. So, θ=ψ̂ because of the uniqueness of ψ̂. Moreover, ρπ=θ=ψ̂=ψ˜π. By the uniqueness of ψ˜, we obtain that ρ=ψ˜.

Example 1.7

  1. Let u=a1a2amRm and v=b1b2bnRn. Define a function ψ:Rm×RnMm×nR given by

    ψuv=utv=a1a2amb1b2bn=a1b1a1b2a1bna2b1a2b2a2bnamb1amb2ambn.E8

    We can see that ψ is a middle linear map. By Theorem 1.6, there exists a unique group homomorphism ψ˜:RmRRnMm×nR such that ψuv=ψ˜uv=utv.

  2. Let A=aijn×nMnR and B=bijn×nMnR. Define a function ψ:MnR×MnRMn2R given by

ψAB=aijB=a11Ba12Ba1nBa21Ba22Ba2nBan1Ban2BannB.E9

We can see that ψ is a middle linear map. By Theorem 1.6, there exists a unique group homomorphism ψ˜:MnRRMnRMn2R such that ψAB=ψ˜AB=aijB.

Let MK, KN, and KU be ternary semimodules over a ternary semifield K. A bilinear map over K from M×N to U is a function T:M×NU such that for all m,m˜M, n,n˜N and α,βK,

  1. Tm+m˜n=Tmn+Tm˜n,

  2. Tmn+n˜=Tmn+Tmn˜, and

  3. Tmαβn=αβTmn=Tmαβn.

The map μ:M×NMKN is given by μmn=mn and is called the canonical bilinear map (over a ternary semifield K).

By Theorem 1.6, the following result is derived.

Corollary 1.8 Let KMK, KN, and KU be ternary semimodules over a ternary semifield K, and μ:M×NMKN the canonical bilinear map. For any bilinear map ψ:M×NU, there exists a unique K-ternary semimodule homomorphism ψ˜:MKNU such that ψ=ψ˜μ, that is, the following diagram commutes.

Example 1.9 We will apply Corollary 1.8 to show that Q0Q0F01F01.

Let μ:Q0×F01Q0Q0F01 be a canonical bilinear map. That is, μaf=af, for all aQ0 and fF01.

Define ψ:Q0×F01F01 by ψaf=1af, where aQ0 and fF01. We can show that ψ is a bilinear map.

By the universal mapping property, there exists a unique ternary homomorphism ψ˜:Q0Q0F01F01 such that ψ=ψ˜μ. That is, ψ˜af=ψ˜μaf=ψaf=1af for all aQ0 and fF01.

We will show that ψ˜ is an isomorphism. Let fF01. We have 1fQ0Q0F01 and ψ˜1f=11f=f. So, ψ˜ is surjective.

Next, we will show that kerψ˜=0. Let afQ0Q0F01. Consider af=11af=11af. Since F01 is a left Q0-ternary semimodule, af=1g for some gF01.

Let 1gkerψ˜. Then, ψ˜1g=0. Thus, g=11g=0, that is, af=10=0. Hence, kerψ˜=0, so ψ˜ is injective. Consequently, ψ˜ is an isomorphism.

By Theorem 1.6, the following results about the tensor products of two ternary semimodule homomorphisms are derived.

Proposition 1.10 Let MK, MK, KN, and KN be ternary semimodules over a ternary semifield K. If f:MM and g:NN are right and left K-ternary semimodule homomorphisms, respectively; then, there exists a unique group homomorphism h from MKN into MKN such that for all mM and nN, hmn=fmgn.

The unique group homomorphism h in Proposition 1.10 is denoted by fg:MKNMKN.

Lemma 1.11 Let K and S be ternary semifields, and SMK and KN be ternary semimodules as indicated. Then, MKN is a left ternary semimodule over S defined by s1s2mn=s1s2mn for all s1, s2S, mM, and nN.

Theorem 1.12 Let K and S be ternary semifields, and SMK, SMK, KN, and KN be ternary semimodules as indicated. If f:MM is a right K-ternary semimodule homomorphism, and g:NN is a left K-ternary semimodule homomorphism; then, fg is a left S-ternary semimodule homomorphism.

Proof: Let f:MM be an SK-ternary bisemimodule homomorphism and g:NN be a left K-ternary semimodule homomorphism. By Lemma 1.11, we obtain that MKN and MKN are left S-ternary semimodules. Moreover, there exists a unique group homomorphism fg:MKNMKN such that fgmn=fmgn for all mM and nN. By Lemma 1.11, we obtain that s1s1i=1pmini=i=1ps1s2mini for all s1,s2S and i=1pminiMKN then

fgs1s1i=1pmini=s1s2fgi=1pmini.E10

Therefore, fg is a left S-ternary semimodule homomorphism.

Proposition 1.13 Let K be a ternary semifield, and MK, MK, MK, KN, KN and KN be ternary semimodules. If f:MM and f:MM are right K-ternary semimodule homomorphisms, g:NN and g:NN are left K-ternary semimodule homomorphisms. Then,

fgfg=ffgg:MKNMKNE11

is a group homomorphism. If f and g are right and left K-ternary semimodule isomorphisms, respectively; then, fg is a group isomorphism and fg1=f1g1 is also a group isomorphism.

Proof: We can see that f˜g˜fg and f˜fg˜g are K-ternary semimodule homomorphisms. It suffices to show that equality holds. Let mM and nN. Then,

f˜g˜fgmn=f˜gfmgn=f˜fmg˜gn=f˜fg˜gmnE12

Hence, fgfg=ffgg. For the last statement, it is straightforward.

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4. The n-fold tensor products

Definition 1.14 Let K be a ternary semifield, and let M1,M2,,Mn and N be appropriate K-ternary semimodules. An n-multilinear map or simply multilinear map (over K) from M1×M2××Mn into N is a function f:M1×M2××MnN such that for all α,βK, miMi and mjMj for all i,j=1,2,n,

  1. fm1mj1mj+mjmj+1mn=fm1mj1mjmj+1mn+fm1mj1mjmj+1mn, and

  2. fm1mj1αβmjmj+1mn=αβfm1mj1mjmj+1mn

We can generalize the tensor products of two ternary semimodules over ternary semifields to tensor products of ternary semimodules M1,M2,,Mn over the same ternary semifield K, which we simply call the n-fold tensor product and write this as M1KM2KKMn by dropping all parentheses.

For example, let n2 and M1,M2,,Mn be ternary semimodules over a ternary semifield K. Then, the function

μn:M1×M2××MnM1KM2KKMn

defined by m1m2mnm1m2mn is a multilinear map. We call the function μn a canonical multilinear map.

Theorem 1.15: (The universal mapping property of the n-fold tensor product) Let n2, and M1,M2,,Mn and N be appropriate ternary semimodule over the same ternary semifield K. For any multilinear map, ψ:M1×M2××MnN, there exists a unique K-ternary semimodule homomorphism ψ˜:M1KM2KKMnN such that ψ=ψ˜μn, i.e., the following diagram commutes.

Proof: Let ψ:M1×M2××Mn×Mn+1N be a multilinear map. For each mMn+1, let ψm:M1×M2××MnN be defined by

ψmm1m2mn=ψm1m2mnmforallmiMi,i=1,2,3,,n.E13

We can see that ψm is a multilinear map. By the induction hypothesis, there exists a unique K-ternary semimodule homomorphism ψ˜m:M1KM2KKMnN such that ψm=ψ˜mμn.

Let B:M1KM2KKMn×Mn+1N be defined by Bxm=ψ˜mx for all x×M1KM2KKMn and mMn+1. Since ψ˜m is unique, B is well-defined. Next, we show that B is a bilinear map.

Let α,βK, x,yM1KM2KKMn and s,tMn+1. Then,

Bx+ys=ψ˜sx+y=ψ˜sx+ψ˜sy=Bxs+Bys,E14

and

Bαβxs=ψ˜sαβx=αβψ˜sx=αβBxs.E15

Moreover, we have to show that Bxs+t=Bxs+Bxt and Bxαβs=αβBxs. That is, ψ˜s+t=ψ˜s+ψ˜t and ψ˜αβs=αβψ˜s, respectively. Because ψ˜s+t is a unique K-ternary semimodule homomorphism such that ψs+t=ψ˜s+tμn; therefore, it suffices to show that ψ˜s+ψ˜tμn=ψs+t. Let miMi for all i=1,2,,n. Then,

ψ˜s+ψ˜tμnm1m2mn=ψ˜sμnm1m2mn+ψ˜tμnm1m2mn=ψsm1m2mn+ψtm1m2mn=ψm1m2mns+ψm1m2mnt=ψm1m2mns+t=ψs+tm1m2mn.E16

Similarly, ψ˜αβs is a unique K-ternary semimodule homomorphism such that ψαβs=ψ˜αβsμn. So, we will show that αβψ˜sμn=ψαβs. Let miMi for all i=1,2,,n. Then,

αβψ˜sμnm1m2mn=αβψ˜sμnm1m2mn=αβψsm1m2mn=αβψm1m2mns=ψm1m2mnαβs=ψαβsm1m2mn.E17

Hence, B is a bilinear map.

For each miMi for all i=1,2,,n,n+1, we consider the bilinear map μ:M1KM2KKMn×Mn+1M1KM2KKMnKMn+1. We can see that

μμnm1m2mnmn+1=μnm1m2mnmn+1=m1m2mnmn+1=m1m2mnmn+1=μn+1m1m2mnmn+1.E18

Fix i1,2,3nn+1 and let α,βK, mi,miMi for all i. If i=1,2,,n. Then,

μn+1m1m2mi+mimnmn+1=μμnm1m2mi+mimnmn+1=μμnm1m2mimn+μnm1m2mimnmn+1=μμnm1m2mimnmn+1+μμnm1m2mimnmn+1=μn+1m1m2mimnmn+1+μn+1m1m2mimnmn+1,E19

and

μn+1m1m2αβmimnmn+1=μμnm1m2αβmimnmn+1=μαβμnm1m2mimnmn+1=αβμμnm1m2mimnmn+1=αβμn+1m1m2mimnmn+1.E20

If i=n+1. Then,

μn+1m1m2mnmn+1+mn+1=μμnm1m2mnmn+1+mn+1=μμnm1m2mnmn+1+μμnm1m2mnmn+1=μn+1m1m2mnmn+1+μn+1m1m2mnmn+1,E21

and

μn+1m1m2mnαβmn+1=μμnm1m2mnαβmn+1=αβμμnm1m2mnmn+1=αβμn+1m1m2mnmn+1.E22

Hence, μn+1 is an (n+1)-multilinear map.

Next, we consider the case n+1, where ψ:M1×M2××Mn+1N is a multilinear map. Because B and μ are bilinear maps and by the induction hypothesis, so there exists a unique K-ternary module homomorphism ψ˜:M1KM2KKMnKMn+1N such that B=ψ˜μ.

Consider the following diagram.

where θ is a function from M1×M2××Mn×Mn+1 into M1KM2KKMn×Mn+1 defined by θm1m2mnmn+1=μnm1m2mnmn+1. To show ψ=ψ˜μn+1, let miMi for all i=1,2,3,,n+1. Then,

ψ˜μn+1m1m2mnmn+1=ψ˜μμnm1m2mnmn+1=Bμnm1m2mnmn+1=ψ˜mn+1μnm1m2mn=ψmn+1m1m2mn=ψm1m2mnmn+1.E23

Thus, ψ=ψ˜μn+1.

Finally, we show that ψ˜ is unique. Suppose that κ:M1KM2KKMnKMn+1N is a K-ternary semimodule homomorphism such that ψ=κμn+1. For each sMn+1, let κs:M1KM2KKMnN be defined by κsx=κμxs for all xM1KM2KKMn. Then, κs is a K-ternary semimodule homomorphism. Let miMi for all i=1,2,3,,n. Then,

κsμnm1m2mn=κμμnm1m2mns=κμn+1m1m2mns=ψm1m2mns=ψsm1m2mn.E24

Thus, κsμn=ψs. By the uniqueness of ψ˜s, we obtain that κs=ψ˜s. Now, we show that B=κμ. Let xM1KM2KKMn and sMn+1. Then, κμxs=κμxs=κsx=ψ˜sx=Bxs=ψ˜μxs. By the uniqueness of ψ˜, we obtain that κ=ψ˜. Hence, ψ˜ is unique.

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5. Exact sequences

For the last section, we investigate short exact sequences of ternary semimodules and flat ternary semimodules. By the way, we follow the definitions of short exact sequences and flat semimodules from Refs. [6, 10] but we change the semimodules to the ternary semimodules.

Theorem 1.16 Let K be a ternary semifield, M be a left K-ternary semimodule and ViiI be a family of right K-ternary semimodules. If each Vi for all iI is M-flat; then, the direct sum Vi is M-flat.

Proof: Let idM be the identity function on a K-ternary semimodule M. Consider the following diagram

where πi:iIViKMViKM given by πivim=vim for all iI and ιi:ViKMiIViKM given by ιivim=vjm where

vjm=vimifi=j,0ifijE25

for all i,jI. There exists a group isomorphism ψ:iIViKMiIViKM such that ψvim=vim.

Assume that Vi is M-flat for each iI. Then,

0=idViidNvin=idViviidNn=vimE26

for some mM. By diagram (*), we have that

0=ψ1ιivim=ψ1vim=vim.E27

So, idiIViidNvin=idiIViviidNn=vim=0.

That is, keridiIViidN=0. Hence, iIVi is M-flat.

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Acknowledgments

We thank the referees for their suggestions and comments on the manuscript.

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Written By

Thorranin Pawaputanon Na Mahasarakham, Ruangvarin Intarawong Sararnrakskul and Nissara Sirasuntorn

Reviewed: 30 November 2022 Published: 29 December 2022