The lattice of ideals of a numerical semigroup and its frobenius restricted variety associated

149 (2024) MATHEMATICA BOHEMICA No. 3, 439–454 THE LATTICE OF IDEALS OF A NUMERICAL SEMIGROUP AND ITS FROBENIUS RESTRICTED VARIETY ASSOCIATED Ma ia Angeles Mo eno-F ías, Pue o Real, José Ca los Rosales, G anada, Recei ed Ma ch 8, 2023. Published online Oc obe 23, 2023. Communica ed by Sándo Radeleczki Abs ac . Le ∆ be a nume ical semig oup. In his wo k we show ha J(∆) = {I∪ {0}:Iis an ideal o ∆}is a dis ibu i e la ice, which in addi ion is a F obenius e- s ic ed a ie y. We gi e an algo i hm which allows us o compu e he se Ja(∆) = {S∈ J(∆): max(∆ S) = a} o a gi en a∈∆.As a consequence, we ob ain ano he algo i hm ha compu es all he elemen s o J(∆) wi h a ixed genus. Keywo ds: nume ical semig oup; ideal; F obenius es ic ed a ie y; embedding dimen- sion; F obenius numbe ; es ic ed F obenius numbe ; genus; mul iplici y; A nume ical semig oup; sa u a ed semig oup MSC 2020: 20M14, 11Y16 1. In oduc ion Le Zbe he se o in ege numbe s and N={x∈Z:x⩾0}. A nume ical semig oup is a subse So N, which is closed by he sum, 0∈Sand N S= {x∈N:x /∈S}is ini e. I Ais a nonemp y subse o N, we deno e by hAi he submonoid o (N,+) gen- e a ed by A, ha is, hAi={λ1a1+...+λnan:n∈N {0},{a1,...,an} ⊆ A and {λ1,...,λn} ⊆ N}.By Lemma 2.1 om [12], we know ha hAiis a nume ical semig oup i and only i gcd(A) = 1. The wo k was pa ially suppo ed by Jun a de Andalucía g oups FQM-298 and FQM-343, P oyec o de Excelencia de la Jun a de Andalucía P oyExcel 00868, P oyec o de in es i- gación del Plan P opio–UCA 2022-2023 (PR2022-011) and P oyec o de in es igación del Plan P opio–UCA 2022-2023 (PR2022-004). c The au ho (s) 2023. This is an open access a icle unde he CC BY-NC-ND licence cbnd DOI: 10.21136/MB.2023.0038-23 439 I Sis a nume ical semig oup and S=hAi, hen we say ha Ais a sys em o gene a o s o S. Mo eo e , i S6=hBi o all B!A, hen we say ha Ais a minimal sys em o gene a o s o S. In [12], Co olla y 2.8, i is shown ha e e y nume ical semig oup has a unique minimal sys em o gene a o s which, in addi ion, is ini e. We deno e by msg(S) he minimal sys em o gene a o s o S. The ca dinali y o msg(S)is called he embedding dimension o Sand is deno ed by ed(S). I Sis a nume ical semig oup, hen F(S) = max(Z S),g(S) = ♯(N S), whe e ♯A deno es he ca dinali y o se A, and m(S) = min(S {0}). They a e h ee impo an in a ian s o Swhich we call F obenius numbe ,genus and mul iplici y o S, espec i ely. Le ∆be a nume ical semig oup. An ideal o ∆is a nonemp y subse Io ∆such ha I+ ∆ = {a+b:a∈Iand b∈∆} ⊆ I. I Iis an ideal o ∆, hen I∪ {0}is a nume ical semig oup. This ac induces us o gi e he ollowing de ini ion. A nume ical semig oup Sis an I(∆)-semig oup i S {0}is an ideal o ∆. We deno e J(∆) = {S:Sis an I(∆)-semig oup}.The main aim o his manusc ip is o s udy he se J(∆). In Sec ion 2, we ecall some basic no ions and esul s o he heo y o ideals o nume ical semig oups. In Sec ion 3, we show ha J(∆) is closed unde union and in e sec ion, and so i is a dis ibu i e la ice. Mo eo e , we show ha i S∈ J (∆) and x= max(∆ S), hen S∪ {x} ∈ J (∆) and consequen ely, we ha e ha J(∆) is a F obenius es ic ed a ie y. We say ha an ideal Io a nume ical semig oup ∆is p incipal i he e exis s a∈∆such ha I={a}+ ∆. A P(∆)-semig oup is a nume ical semig oup wi h he o m ({a}+∆)∪{0}and a∈∆.In Sec ion 4, we illus a e ha e e y I(∆)-semig oup can be exp essed as a ini e and i edundan union o P(∆)-semig oups. By P oposi ion 2.10 om [12], we know ha i Sis a nume ical semig oup, hen ed(S) ⩽m(S).A MED-semig oup is a nume ical semig oup Ssuch ha ed(S) = m(S).This class o nume ical semig oups has been widely s udied, see o ins ance [3]. In Sec ion 4, we show ha i Sis an P(∆)-semig oup and S6= ∆, hen Sis a MED- semig oup. Inspi ed by [1], Lipman in oduces and mo i a es in [6] he s udy o A ings. The cha ac e iza ion o hese ings ia hei alue semig oups yields he no ion o A nume ical semig oup. E e y A nume ical semig oup is a MED-semig oup. In Sec ion 4, we show ha ∆is an A nume ical semig oup i and only i e e y P(∆)- semig oup is an A nume ical semig oup. A pa icula ly in e es ing ype o nume ical semig oups a e called sa u a ed nu- me ical semig oups. The idea o sa u a ion o singula i ies we e in oduced in h ee di e en ways by: Za iski in [13]–[15], Pham-Teissie in [9], and Campillo in [4]. As o he A p ope y, sa u a ed nume ical semig oups come in o scene a e a cha - 440 ac e iza ion o sa u a ed ings in e ms o hei alue semig oups (see [5], [8]). In Sec ion 4, we show ha a nume ical semig oup ∆is sa u ed i and only i he e is a leas one P(∆)-sa u ed semig oup. Le ∆be a nume ical semig oup. We say ha an ideal is i educible i i canno be exp essed as he in e sec ion o wo ideals p ope ly con aining i . I a∈∆, hen we deno e B(a) = {s∈∆: a−s∈∆}.As a consequence o Lemma 3.1 om [2] we ha e ha Iis an i educible ideal o ∆i and only i I= ∆ o I= ∆ B(a) o some a∈∆.AD(∆)-semig oup is a nume ical semig oup wi h he o m (∆ B(a)) ∪ {0} o some a∈∆.As a consequence o Theo em 3.3 om [2], we ha e ha e e y I(∆)-semig oup can be exp essed as a unique, ini e and i edundan in e sec ion o D(∆)-semig oups. I S(Ta e nume ical semig oups, hen he F obenius numbe o S es ic ed o Tis FT(S) = max(T S).I ∆is a nume ical semig oup and a∈∆, hen we pu Ja(∆) = {S:Sis an I(∆)-semig oup and F∆(S) = a}.In Sec ion 5, we o de he elemen s o he se Ja(∆) in he o m o a ee wi h oo (∆ B(a)) ∪ {0}.This ac allows us in Sec ion 6 o gi e an algo i hm which compu es all he elemen s o he se Ja(∆).Finally and based on he p e ious algo i hm, we show ano he one, o di e en na u e and wi h complexi y no compa able o he algo i hm p esen ed in [7], ha allows us o compu e he se J(∆, k) = {S:Sis an I(∆)-semig oup and g(∆) = g(∆) + k} o all k∈N. 2. Basic concep s and esul s Le ∆be a nume ical semig oup. An ideal o ∆is a nonemp y subse Io ∆such ha I+ ∆ ⊆I. The ollowing esul has an easy p oo . P oposi ion 2.1. I Iand Ja e ideals o a nume ical semig oup ∆, hen I∪J and I∩Ja e also ideals o ∆. I is clea ha i Iis an ideal o ∆, hen ∆ Iis ini e. The e o e, i I6= ∆, hen he e exis s max(∆ I). P oposi ion 2.2. Le ∆be a nume ical semig oup, le Ibe an ideal o ∆such ha I6= ∆ and x= max(∆ I).Then I∪ {x}is an ideal o ∆. P o o . By maximili y o x, we ha e {x}+∆ ⊆I∪{x}.The e o e, (I∪{x})+∆ ⊆ I∪ {x}. The ollowing esul is P oposi ion 1 om [7]. P oposi ion 2.3. I ∆is a nume ical semig oup and Xis a nonemp y subse o ∆, hen X+ ∆ is an ideal o ∆.Mo eo e , e e y ideal o ∆has his o m. 441 I ∆is a nume ical semig oup, hen we de ine o e Z he ollowing o de ela ion: a⩽∆bi and only i b−a∈∆.We say ha a nonemp y subse Xo ∆is a ∆- incompa able se i a−b /∈∆ o all (a, b)∈X×Xsuch ha a6=b. The ollowing esul is Theo em 5 om [7]. Theo em 2.4. Le ∆be a nume ical semig oup. Then he se {X+ ∆: Xis a ∆-incompa able se } is he se o med by all he ideals o S. Mo eo e , i Xand Ya e di e en ∆- incompa able se s, hen X+ ∆ 6=Y+ ∆. I Iis an ideal o a nume ical semig oup ∆and I=X+ ∆, hen we say ha X is an ideal sys em o gene a o s o I. Mo eo e , i Xis a ∆-incompa able se , hen we say ha Xis he ideal minimal sys em o gene a o s o I. By Theo em 2.4, we know ha e e y ideal Io ∆admi s a unique ideal minimal sys em o gene a o s. We deno e his sys em by imsg∆(I). The ollowing esul is P oposi ion 2.6 om [7]. P oposi ion 2.5. Le ∆be a nume ical semig oup and le Ibe an ideal o ∆. Then imsg∆(I) = Minimals⩽∆(I). The ollowing esul is P oposi ion 7 om [7]. P oposi ion 2.6. I ∆is a nume ical semig oup and le Xbe a ∆-incompa able se , hen Xis ini e. As a consequence o P oposi ions 2.5 and 2.6, he ca dinal o imsg∆(I)is an in ege posi i e numbe . This numbe is called he ideal dimension o Iin ∆and i is deno ed by dim∆(I). The ollowing esul is P oposi ion 8 om [7]. P oposi ion 2.7. I Iis an ideal o ∆, hen: (1) I= ∆ i and only i 0∈I, (2) I∪ {0}is a nume ical semig oup. The ollowing esul is P oposi ion 9 om [7]. P oposi ion 2.8. Le ∆be a nume ical semig oup and le Ibe an ideal o ∆ such ha I6= ∆. Then imsg∆(I) = Minimals⩽∆(msg(I ∪ {0})). As an immedia e consequence o P oposi ion 2.8, we ha e he ollowing esul . Co olla y 2.9. Le ∆be a nume ical semig oup and le Ibe an ideal o ∆. Then dim∆(I)⩽ed(I ∪ {0}). 442 I is well known ha i ∆is a nume ical semig oup and x∈∆, hen ∆ {x}is a nume ical semig oup i and only i x∈msg(∆). The ollowing esul is easy o p o e. P oposi ion 2.10. I Iis an ideal o ∆and x∈I, hen I {x}is an ideal o ∆ i and only i x∈imsg∆(I). 3. I(∆)-semig oups Le ∆be a nume ical semig oup. By P oposi ion 2.7, we know ha i Iis an ideal o ∆, hen I∪ {0}is a nume ical semig oup. An I(∆)-semig oup is a nume ical semig oup Ssuch ha S {0}is an ideal o ∆.We pu J(∆) = {S:Sis an I(∆)-semig oup}. E x a m p l e 3.1. I is clea ha Xis an N-incompa able se i and only i X={n} o e e y n∈N.Hence, by applying Theo em 2.4, J(N) = {{0, n, →}:n∈N}( he symbol →means ha e e y in ege g ea e han nbelongs o he se ). The nume ical semig oups wi h he o m {0, n, →} a e called o dina y nume ical semig oups. So he concep s o I(N)-semig oup and o dina y nume ical semig oup a e equi alen . I Sand Ta e nume ical semig oups and S⊆T, he F obenius numbe o S es ic ed o Tis FT(S) = max(T S).By de ini ion FT(T) = −1. By applying P oposi ions 2.1 and 2.2, we can easily deduce he ollowing esul . Theo em 3.2. Le ∆be a nume ical semig oup. Then: (1) I {S, T } ⊆ J (∆), hen {S∪T, S ∩T} ⊆ J (∆). (2) ∆is he maximum elemen (wi h espec o se inclusion) o J(∆). (3) I S∈ J (∆) and S6= ∆, hen S∪ {F∆(S)} ∈ J (∆). Ala ice is an algeb aic s uc u e (L, ∨,∧)consis ing o a se Land wo bina y ope a ions ∨and ∧o e Lsa is ying he p ope ies: commu a i e, associa i e, idem- po en and abso p ion. I , in addi ion, i e i ies he dis ibu i e p ope y, hen he la ice is called dis ibu i e. As an immedia e consequence o Theo em 3.2, we ha e he ollowing esul . Co olla y 3.3. I ∆is a nume ical semig oup, hen (J(∆),∪,∩)is a dis ibu i e la ice. AF obenius es ic ed a ie y (see [10]) is a nonemp y amily Fo nume ical semig oups e i ying he ollowing condi ions: 443 (1) Fhas a maximum elemen (and we deno e i ∆(F)). (2) I {S, T } ⊆ F, hen S∩T∈ F. (3) I S∈ F and S6= ∆(F), hen S∪ {F∆(F)(S)} ∈ F. As an immedia e consequence o Theo em 3.2, we ha e he ollowing esul . Co olla y 3.4. I ∆is a nume ical semig oup, hen J(∆) is a F obenius e- s ic ed a ie y. 4. P(∆)-semig oups In he es o his wo k ∆deno es a nume ical semig oup. An ideal Io ∆is p incipal i dim∆(I) = 1.So he se o med by all he p incipal ideals o ∆is {{a}+ ∆: a∈∆}. P oposi ion 4.1. I Iis an ideal o ∆, hen he nex condi ions a e equi alen : (1) Iis a p incipal ideal. (2) Icanno be exp essed as he union o wo ideals o ∆s ic ly con ained in I. P o o . (1) ⇒(2): Le Jand Kbe ideals o ∆such ha J⊆I, K ⊆I and I=J∪K. As Iis a p incipal ideal o ∆, hen he e exi s a∈∆such ha I={a}+ ∆.Then a∈I=J∪Kand hence a∈Jo a∈K. I a∈J, hen I={a}+ ∆ ⊆J+ ∆ ⊆Jand so I=J. (2) ⇒(1): I Iis no a p incipal ideal o ∆, hen dim∆(I) = n⩾2.The e o e, he e exis s {a1, a2,...,an}a∆-incompa able se such ha {a1,...,an}+ ∆ = I. Le J={a1}+ ∆ and K={a2,...,an}+ ∆.Then Jand Ka e ideals o ∆such ha J⊆I, K ⊆Iand I=J∪K. Mo eo e , applying ha {a1, a2,...,an}is a ∆-incompa able se , we deduce ha J(Iand K(I.  AP(∆)-semig oup is a nume ical semig oup wi h he shape ({a}+ ∆) ∪ {0} o some a∈∆.We pu P(∆) = {S:Sis a P(∆)-semig oup}. P oposi ion 4.2. Le ∆be a nume ical semig oup. (1) I {S1, S2,...,Sn} ⊆ P(∆), hen S1∪S2∪...∪Sn∈ J (∆). (2) I S∈ J (∆) and dim∆(S {0}) = n, hen he e exis s {S1, S2,...,Sn} ⊆ P(∆) such ha S=S1∪S2∪. . . ∪Sn. P o o . (1) I is a consequence om Theo em 3.2. (2) I dim∆(S {0}) = n, hen he e exis s {x1, x2,...,xn} ⊆ ∆such ha S {0}= {x1,...,xn}+ ∆.Fo e e y i∈ {1,...,n}, le Si= ({xi}+ ∆) ∪ {0}.I is clea ha Si∈ P(∆) o all i∈ {1,...,n}and S=S1∪...∪Sn. 444 We say ha a union S i∈{1,...,n} Aio he se s Aiis i edundan i o e e y j∈ {1,...,n},i is e i ied ha S i∈{1,...,n} Ai6=S i∈{1,...,n} {j} Ai.The ollowing esul has an easy p oo . P oposi ion 4.3. E e y I(∆)-semig oup can be exp essed in a unique way as a ini e and i edundan union o P(∆)-semig oups. The ollowing esul is deduced om [11], P oposi ion 2. P oposi ion 4.4. I Sis a P(∆)-semig oup and S6= ∆, hen Sis a MED- semig oup. The ollowing esul can be easily deduced om [11], P oposi ion 9. P oposi ion 4.5. I ∆6=N, a ∈∆ {0}and S= ({a}+ ∆) ∪ {0}, hen F(S) = a+ F(∆),g(S) = a−1 + g(∆) and m(S) = a. As an immedia e consequence o he p e ious p oposi ion we ha e he ollowing esul . Co olla y 4.6. I {S, T } ⊆ P(∆), hen he ollowing condi ions a e equi alen : (1) S=T, (2) m(S) = m(T), (3) F(S) = F(T), (4) g(S) = g(T). No e ha as a consequence o P oposi ion 4.5 and Co olla y 4.6, he numbe o elemen s o P(∆) wi h F obenius numbe F, genus g, mul iplici y m, espec i ely, is 1 o 0depending on whe he he e exis s a∈∆such ha F = F(∆)+a, g = g(∆)+a−1, m = a, espec i ely. A nume ical semig oup Sis A i x+y−z∈S o e e y x, y, z ∈Ssuch ha z⩽y⩽x. I Sis an A nume ical semig oup, hen by [12], P oposi ion 3.12, we can deduce ha Sis a MED-semig oup. The ollowing esul ollows om [11], Co olla y 38. P oposi ion 4.7. ∆is an A nume ical semig oup i and only i all he elemen s o he se P(∆) a e A nume ical semig oups. I A⊆Nand a∈A {0}, hen we deno e dA(a) = gcd{x∈A:x⩽a}.A nume ical semig oup is sa u a ed i s+ dS(s)∈S o all s∈S {0}. By Lemma 3.31 om [12], we know ha e e y sa u a ed nume ical semig oup is an A nume ical semig oup. The ollowing esul is deduced om [11], Co olla y 43. 445 P oposi ion 4.8. ∆is a sa u a ed nume ical semig oup i and only i P(∆) {∆} con ains a leas a sa u a ed nume ical semig oup. 5. D(∆)-semig oups Le ∆be a nume ical semig oup. An ideal is i educible i i canno be exp essed as he in e sec ion o wo ideals p ope ly con aining i . I a∈∆, hen we deno e B(a) = {s∈∆: a−s∈∆}.The ollowing esul ollows om [2], Lemma 3.1. P oposi ion 5.1. Iis an i educible ideal o ∆i and only i I= ∆ B(a) o some a∈∆o I= ∆. AD(∆)-semig oup is a nume ical semig oup wi h he o m (∆ B(a))∪{0} o some a∈∆.We pu D(∆) = {S:Sis a D(∆)-semig oup}.We say ha an in e sec ion T i∈{1,...,n} Aio he se s Aiis i edundan i T i∈{1,...,n} Ai6=T i∈{1,...,n} {j} Ai o e e y j∈ {1,...,n}. The ollowing esul is deduced om [2], Theo em 3.3. P oposi ion 5.2. E e y I(∆)-semig oup can be exp essed as a unique ini e and i edundan in e sec ion o D(∆)-semig oups. I ∆is a nume ical semig oup and a∈∆, hen we pu S(∆, a) = (∆ B(a)) ∪ {0} ∈ D(∆). The ollowing esul has an easy p oo . P oposi ion 5.3. I ∆is a nume ical semig oup and a∈∆ {0}, hen F∆(S(∆, a)) = aand g(S(∆, a)) = g(∆) + #B(a)−1. R e m a k 5.4. ⊲Obse e ha S(∆,0) = ∆ and so F∆(S(∆,0)) = F∆(∆) = −1.The e o e, {F∆(S(∆, a)): a∈∆}= (∆ {0})∪ {−1}. ⊲We p opose he s udy o he se {#B(a): a∈∆}as an open p oblem. Theo em 5.5. Le Sbe a nume ical semig oup. Then Sis a D(∆)-semig oup i and only i Sis a maximal elemen (wi h espec o se inclusion) o he se {T:is an I(∆)-semig oup and F∆(T) = F∆(S)}. P o o . Necessi y. I Sis no maximal, hen he e exis s an I(∆)-semig oup T such ha S(Tand F∆(T) = F∆(S).By Theo em 3.2, we know ha S∪ {F∆(S)} is an I(∆)-semig oup. Then S= (S∪ {F∆(S)})∩Tand so we ha e been able o w i e Sas an in e sec ion o wo I(∆)-semig oups p ope ly con aining S. Hence, S is no a D(∆)-semig oup. 446 Su iciency. Le T= (∆ B(F∆(S)))∪{0}.I is clea ha Tis an I(∆)-semig oup, S⊆Tand F∆(T) = F∆(S).By applying he maximili y o S, we ob ain ha S=T. The e o e, Sis a D(∆)-semig oup.  As a consequence o P oposi ions 5.1 and 5.3, we deduce he ollowing esul . P oposi ion 5.6. I Sis an I(∆)-semig oup, hen he e exis s a unique D(∆)- semig oup Tsuch ha S⊆Tand F∆(T) = F∆(S).Mo eo e , T= S(∆,F∆(S)) i S6= ∆ and T= ∆ i S= ∆. I ∆is a nume ical semig oup and a∈∆ {0}, hen we pu Ja(∆) = {S: Sis an I(∆)-semig oup and F∆(S) = a}. P oposi ion 5.7. Le Sbe an I(∆)-semig oup such ha S6= ∆ and F∆(S) = a. Then S= S(∆, a)i and only i {h∈∆ S:h /∈B(a)}=∅. P o o . I S= S(∆, a), hen S= (∆ B(a)) ∪ {0}and so {h∈∆ S: h /∈B(a)}=∅.Con e sely, i {h∈∆ S:h /∈B(a)}=∅, hen i is clea ha S= S(∆, a). I S∈ Ja(∆) and S6=S(∆, a), hen we pu α(S) = max{h∈∆ S:h /∈B(a)}. By de ini ion, α((S, a)) = 0. P oposi ion 5.8. I a∈∆and S∈ Ja(∆), hen S∪ {α(S)} ∈ Ja(∆). P o o . By he maximali y o α(S),we deduce ha {α(S)}+ ∆ ⊆S∪ {α(S)}. F om his esul one easily deduces ha S∪ {α(S)} ∈ Ja(∆). I S∈ Ja(∆), hen he p e ious p oposi ion can be used o de ine ecu si ely he ollowing sequence o elemen s o Ja(∆): ⊲ S0=S, ⊲ Sn+1 =Sn∪ {α(Sn)} o all n∈N. As a consequence o P oposi ions 5.7 and 5.8, we ha e he ollowing esul . P oposi ion 5.9. I a∈∆and S∈ Ja(∆), hen he e is p∈Nsuch ha S=S0(S1(...(Sp=S(∆, a). Ag aph Gis a pai (V, E)whe e Vis a nonemp y se and E⊆ {(u, )∈V×V: u6= }. The elemen s o Vand Ea e called e ices and edges, espec i ely. A pa h o leng h nconnec ing he e ices xand yo g aph Gis a sequence o di e en edges o he o m ( 0, 1),( 1, 2),...,( n−1, n)such ha 0=xand n=y. A g aph is a ee i G= (V, E),whe e he e exis s a e ex (known as he oo o G) such ha o any o he e ex xo G he e exis s a unique pa h connec ing x and . I (u, )is an edge o a ee, hen we say ha uis a child o . 447 [15] O. Za iski: Gene al heo y o sa u a ion and o sa u a ed local ings III. Sa u a ion in a bi a y dimension and, in pa icula , sa u a ion o algeb oid hype su aces. Am. J. Ma h. 97 (1975), 415–502. zbl MR doi Au ho s’ add esses:Ma ia Angeles Mo eno-F ías (co esponding au ho ), Dp o. de Ma emá icas, Facul ad de Ciencias, Uni e sidad de Cádiz, E-11510, Pue o Real, Cádiz, Spain, e-mail: ma iangeles.mo [email protected];José Ca los Rosales, Dp o. de Álgeb a, Fac- ul ad de Ciencias, Uni e sidad de G anada, E-18071, G anada, Spain, e-mail: j osales@ ug .es. 454