Closure Operations and Nuclei

Abstract

This chapter generalizes many of the requisite facts on semistar, star, and semiprime operations on commutative rings employed in Chapters 1–4 to nuclei on ordered magmas satisfying various algebro-order-theoretic hypotheses. This serves to situate the theories of semistar, star, and semiprime operations on commutative rings in the broader contexts of closure operations, order theory, and the theories of ordered monoids, quantales, and multiplicative lattices.

Exercises

Section 6.1

• 1. Prove that a poset S is complete (resp., near sup-complete, bounded complete) if and only if $\inf _S X$ exists for every subset (resp., every bounded below subset, every nonempty bounded below subset) X of S.

• 2. Prove that a closure operation on a poset S is equivalently a self-map $\star$ of S such that $x \leqslant y^\star$ if and only if $x^\star \leqslant y^\star$ for all $x,y \in S$.

• 3. Prove Lemma 6.1.6.

• 4. Prove Proposition 6.1.11.

• 5. Verify Example 6.1.21.

• 6. Prove Proposition 6.1.15.

• 7. Verify (1), (3), and (5) of Example 6.1.19.

• 8. Verify (6) of Example 6.1.19.

• 9. Verify (2) of Example 6.1.19.

• 10. Prove Proposition 6.1.22.

• 11. Verify Remark 6.1.23.

• 12. Prove statement (1) of Proposition 6.1.26.

• 13. Prove Corollary 6.1.27.

  ordered magma M	abbr.	$\sup X$ exists	$\sup (XY) = \sup ( X) \sup (Y)$
  ordered magma	om
  sup-magma	s	for all X
  near sup-magma	ns	for all $X \ne \varnothing$
  dcpo magma	d	for all directed X
  bounded complete	bc	for all $X \ne \varnothing$ bounded above
  bounded above	b	for $X = M$
  with annihilator	a	for $X = \varnothing$	for $X = \varnothing$ or $Y = \varnothing$
  prequantale	p	for all X	for all X, Y
  near prequantale	np	for all $X \ne \varnothing$	for all $X,Y \ne \varnothing$
  semiprequantale	sp	for all finite or bounded $X \ne \varnothing$	for all (finite or) bounded $X,Y \ne \varnothing$
  prequantic semilattice	ps	for all finite X	for all finite X, Y
  multiplicative semilattice	ms	for all finite $X \ne \varnothing$	for all finite $X,Y \ne \varnothing$
  Scott-topological	t		for all directed X, Y such that $\sup X$, $\sup Y$ exist
  residuated	r	if $\exists x,y : \ X =$ $\{z: zy \leqslant x\}$ or $X = \{z: yz \leqslant x\}$	for all X, Y such that $\sup X$, $\sup Y$ exist
  near residuated	nr	if $X \ne \varnothing$ and $\exists x,y :$ $X = \{z: zy \leqslant x\}$ or $X = \{z: yz \leqslant x\}$	for all $X,Y \ne \varnothing$ such that $\sup X$, $\sup Y$ exist

• 14. Equivalent characterizations of various classes of ordered magmas are given in the table above. Show that the three lattices in Figure 6.1 are each full lattices of implications.

• 15. Let M be an ordered magma. Prove the following.

  • a) If M is unital and residuated, or more generally if M is left or right unital and for all $a \in M$ the left and right multiplication by a maps on M are sup-preserving, then $\Sigma$ is a sup-spanning subset of M if and only if every element of M is the supremum of some subset of $\Sigma$.

  • b) Let $\star$ be a closure operation on an ordered magma M and let $\Sigma$ be any sup-spanning subset of M. Then $\star$ is a nucleus on M if and only if $ax^\star \leqslant (ax)^\star$ and $x^\star a \leqslant (xa)^\star$ for all $a \in \Sigma$ and all $x \in M$, and in that case $\Sigma ^\star$ is a sup-spanning subset of $M^\star$.

  • c) The set ${\text {Prin}}(D)$ of all nonzero principal fractional ideals of an integral domain D is a sup-spanning subset of the residuated near multiplicative lattice $\mathbf {F}^{{{\text {reg}}}}(D)$.

  • d) A nucleus on $\mathbf {F}^{{{\text {reg}}}}(D)$ is equivalently a closure operation $*$ on $\mathbf {F}^{{{\text {reg}}}}(D)$ such that $(aI)^* = aI^*$ for all $I \in \mathbf {F}^{{{\text {reg}}}}(D)$ and all nonzero $a \in T(D)$.

• 16. For any self-map $\star$ of a magma M, we say that $a \in M$ is transportable through $\star$ if $(ax)^\star = ax^\star$ and $(xa)^\star = x^\star a$ for all $x \in M$. We let ${\mathsf T}^\star (M)$ denote the set of all elements of M that are transportable through $\star$.

  • a) Show that any closure operation $\star$ on an ordered magma M such that ${\mathsf T}^\star (M)$ is a sup-spanning subset of M is a nucleus on M.

  • b) For any magma M we let ${\textsf {U }}(M)$ denote the set of all $u \in M$ for which there exists $u^{-1} \in M$ such that the left and right multiplication by $u^{-1}$ maps are inverses, respectively, to the left and right multiplication by u maps. Show that ${\textsf {U }}(M) \subseteq {\mathsf T}^\star (M)$ for any nucleus $\star$ on an ordered magma M.

• 17. A weak ideal system on a commutative monoid M with annihilator 0 is a closure operation r on the multiplicative lattice $2^{M}$ such that $0 \in \varnothing ^r$, $cM \subseteq \{c\}^r$, and $cX^r \subseteq (cX)^r$ for all $c \in M$ and all $X \in 2^{M}$ [95]. A weak ideal system on M is said to be an ideal system on M if $(cX)^r = cX^r$ for all such c and X. Let M be a commutative monoid with annihilator 0. Prove the following.

  • a) A weak ideal system on M is equivalently a nucleus r on the multiplicative lattice $2^{M}$ such that $\{0\}^r = \varnothing ^r$ and $\{1\}^r = M$.

  • b) A weak ideal system r on M is an ideal system on M if and only if every singleton in $2^{M}$ is transportable through r, in the sense of the previous exercise.

  • c) Let R be a commutative ring. A semiprime operation may be seen equivalently as a weak ideal system on the commutative monoid $R^{\bullet }$ of R under multiplication.

  • d) $(**)$ Generalize the notion of a weak ideal system to accommodate the left semiprime operations on a (possibly noncommutative) ring.

• 18. For any magma M, let $M_0$ denote the magma $M \amalg \{0\}$, where $0x = 0 = x0$ for all $x \in M_0$ is an annihilator of $M_0$. Let G be an abelian group. A module system on G is a closure operation r on the multiplicative lattice $2^{G_0}$ such that $\varnothing ^r = \{0\}$ and $(cX)^r = cX^r$ for all $c \in G_0$ and all $X \in 2^{G_0}$ [96]. Prove the following.

  • a) A module system on G is equivalently a nucleus r on the multiplicative lattice $2^{G_0}$ such that $\varnothing ^r = \{0\}$.

  • b) The following are equivalent for any self-map r of $2^{G_0}$ such that $\varnothing ^r = \{0\}$.

    • 1) r is a module system on G.

    • 2) r is a closure operation on the poset $2^{G_0}$ and r-multiplication on $2^{G_0}$ is associative.

    • 3) r is a closure operation on the poset $2^{G_0}$ and $(X^r Y^r)^r = (XY)^r$ for all $X,Y \in 2^{G_0}$.

    • 4) $XY \subseteq Z^r$ if and only if $XY^r \subseteq Z^r$ for all $X,Y,Z \subseteq G_0$.

  • c) $2^{G_0} {-}2^G$ is a sub multiplicative lattice of $2^{G_0}$, and a module system on G may be seen equivalently as a nucleus on the multiplicative lattice $2^{G_0} {-}2^G$.

  • d) Let D be an integral domain, $K^\times$ the group of nonzero elements of the quotient field K of D, and $\star$ a self-map of $\mathbf {K}(D)$. Let $\varnothing ^r = \{0\}^ r = \{0\}$ and $X^r = (DX)^\star$ for all subsets X of K containing a nonzero element. Then $\star$ is a semistar operation on D if and only if r is a module system on the abelian group $K^\times$.

  • e) $(**)$ Generalize the notion of a module system in order to accommodate semistar operations on commutative rings.

• 19. A morphism $f: N \longrightarrow M$ of complete ordered magmas is a sup-preserving map from N to M such that $f(ab) = f(a)f(b)$ for all $a,b \in N$. Let $f: Q \rightarrow M$ be a morphism of complete ordered magmas, where Q is a prequantale. Prove the following.

  • a) If $\star$ is any nucleus on Q, then the corestriction ${_{Q^\star }}|\star : Q \longrightarrow Q^\star$ of $\star$ to $Q^\star$ is a surjective morphism of prequantales.

  • b) There exists a unique nucleus $\star$ on Q such that $f = (f|_{Q^\star }) \circ ({_{Q^\star }}|\star )$ and $f|_{Q^\star }$ is injective; moreover, one has $x^\star = \sup \{y \in Q: \ f(y) = f(x)\}$ for all $x \in Q$.

  • c) $f|_{Q^\star }: Q^\star \longrightarrow M$ is an embedding of complete ordered magmas.

  • d) $f|_{Q^\star }: Q^\star \longrightarrow {\text {im}} f$ is an isomorphism of ordered magmas.

  • e) A quantale Q is said to be simple if every nontrivial sup-preserving semigroup homomorphism from Q is injective. A quantale Q is simple if and only if $d: x \longmapsto x$ and $e: x \longmapsto \sup Q$ are the only nuclei on Q.

  • f) A multiplicative lattice Q is simple if and only if $Q = \{0,1\}$, where $1 = \sup Q$ and 0 is the identity element of Q and $0 = \inf Q$ is an annihilator of Q (and possibly $0 = 1$).

• 20. State and prove a generalization of the previous exercise for a (near sup-preserving) morphism from a near prequantale to a near sup-complete ordered magma.

Section 6.2

• 1. Fill in the missing details in the proof of Proposition 6.2.2.

• 2. Prove statement (2) of Proposition 6.2.4.

• 3. Verify Remark 6.2.8.

• 4. Prove statement (1) of Proposition 6.2.9.

• 5. Use Proposition 6.2.9 to prove statements (2) and (3) of Proposition 4.1.20.

• 6. Verify Remark 6.2.10.

• 7. State and prove analogues of all the results of this section for the poset ${\textsf {Nucl }}_l(M)$ of all left nuclei on an ordered magma M.

• 8.

  • a) Let $\star$ be a closure operation on a residuated semiquantale Q. For all $x \in Q$, let

    $$x^{\lceil \star \rceil } = \inf \{x,\inf \{y \in Q: y \geqslant x \text{ and } \forall z,w \in Q \ (zw \leqslant y \Rightarrow z^\star w^\star \leqslant y)\}\}.$$

    Show that $\lceil \star \rceil : x \longmapsto x^{\lceil \star \rceil }$ is the smallest nucleus on Q larger than $\star$. Also show that, for all $x \in Q$, one has $x^{\lceil \star \rceil } = x$ if and only if $zw \leqslant x$ implies $z^\star w^\star \leqslant x$ for all $z,w \in Q$.

  • b) State and prove an analogous result for left nuclei.

Section 6.3

• 1. Prove Proposition 6.3.3.

• 2. Prove Proposition 6.3.4.

• 3. Prove Proposition 6.3.6 and Corollary 6.3.7.

• 4. Let $\star$ be a closure operation on a near prequantale Q. Show that $v(Q^\star )$ is the largest nucleus on Q smaller than $\star$.

• 5. Prove Proposition 6.3.8.

• 6. Prove statement (3) of Theorem 6.3.11.

• 7. For any ordered magma M, let ${\textsf {U }}(M)$ denote the set of all $u \in M$ such that the left and right multiplication by u maps are poset automorphisms of M. We say that a ${\textsf {U }}$ -lattice (resp., near ${\textsf {U }}$ -lattice, semi-${\textsf {U }}$ -lattice) is an ordered magma M that is complete (resp., near sup-complete, a bounded complete join semilattice) and such that ${\textsf {U }}(M)$ is a sup-spanning subset of M. Prove the following.

  • a) For any commutative ring R, one has ${\textsf {U }}(\mathbf {K}(R)) = {\textsf {U }}(\mathbf {F}^{{\text {reg}}}(R)) = {\text {Inv}}(R)$.

  • b) For any commutative ring R, the ordered monoid $\mathbf {F}^{{\text {reg}}}(R)$ is a semi-${\textsf {U }}$-lattice if and only if R is quasi-Marot.

  • c) For any integral domain D, the multiplicative lattice $\mathbf {K}(D)$ is a ${\textsf {U }}$-lattice.

  • d) Any ${\textsf {U }}$-lattice (resp., near ${\textsf {U }}$-lattice, semi-${\textsf {U }}$-lattice) is a prequantale (resp., near prequantale, semiprequantale).

  • e) Let $\star$ be a nucleus on an ordered magma M. If M is a ${\textsf {U }}$-lattice (resp., near ${\textsf {U }}$-lattice, semi-${\textsf {U }}$-lattice), then so is $M^\star$.

  • f) Let Q be an associative unital semi-${\textsf {U }}$-lattice, and let $a \in Q$. Then v(a) exists and $x^{v(a)} = \inf \{uav: \ u,v \in {\textsf {U }}(Q) \text{ and } x \leqslant uav\}$ for all $x \in Q$.

  • g) Let D be an integral domain, and let $J \in \mathbf {K}(D)$. Then $I^{{\pmb v}(J)} = \bigcap \{HJ: \ H \in {\text {Inv}}(D) \text{ and } I \subseteq HJ\}$ for all $I \in \mathbf {K}(D)$.

  • h) Let R be a quasi-Marot ring, and let $J \in \mathbf {F}^{{\text {reg}}}(R)$. Then $I^{{\pmb v}(J)} = \bigcap \{HJ: \ H \in {\text {Inv}}(R) \text{ and } I \subseteq HJ\}$ for all $I \in \mathbf {F}^{{\text {reg}}}(R)$.

• 8. Let M be any magma. For any $r \in M$, define self-maps $L_r$ and $R_r$ of M by $L_r(x) = rx$ and $R_r(x) = xr$ for all $x \in M$, which we call translations. Let ${\text {Lin}}(M)$ denote the submonoid of the monoid of all self-maps of M generated by the translations. Prove the following.

  • a) If M is a monoid, then any element of ${\text {Lin}}(M)$ can be written in the form $L_r \circ R_s = R_s \circ L_r$ for some $r,s \in M$.

  • b) Let Q be a near prequantale and $a \in Q$. One has

    $$x^{v(a)} = \sup \{y \in Q: \forall f \in {\text {Lin}}(Q) \, (f(x) \leqslant a \Rightarrow f(y) \leqslant a)\}$$

    for all $x \in Q$. Alternatively, $x^{v(a)}$ for all $x \in Q$ is the largest element of Q such that $f(x) \leqslant a$ implies $f(x^{v(a)}) \leqslant a$ for all $f \in {\text {Lin}}(Q)$.

Section 6.4

• 1. Verify the claims made in Definitions 6.4.3 and 6.4.4.

• 2.

  • a) Prove Lemma 6.4.6 and Corollary 6.4.7.

  • b) Show that a semistar, star, or semiprime operation is of finite type if and only if it is finitary as a map of posets.

• 3. Given an example of an infinite poset all of whose elements are compact and an example of an infinite poset having no compact elements.

• 4. Verify Example 6.4.12.

• 5. Consider the following properties of maps between posets.

  • 1) sup-preserving.

  • 2) near sup-preserving.

  • 3) Scott continuous.

  • 4) order-preserving.

    • a) Show that $(1) \Rightarrow (2) \Rightarrow (3) \Rightarrow (4)$.

    • b) Which of the four properties are preserved under composition of functions? Prove your answers correct.

• 6. A subset I of a poset S is said to be an ideal of S if I is a directed downward closed subset of S. For any $x \in S$, the set

  $$\downarrow \!\! x= \{y \in S: \ y \leqslant x\}$$

  is an ideal of S called the principal ideal generated by x. The ideal completion ${\mathsf {Idl }}(S)$ of S is the set of all ideals of S partially ordered by the subset relation. If S is a join semilattice, then, for any nonempty subset X of S, we let

  $$\downarrow \!\! X = \{y \in S: \ y \leqslant \sup T \text{ for } \text{ some } \text{ finite } \text{ nonempty } T \subseteq X\}.$$

  Prove the following.

  • a) Let S be a join semilattice.

    • i) For any nonempty subset X of S, the set $\downarrow \!\! X$ is the smallest ideal of S containing X.

    • ii) The operation $\downarrow$ is a finitary closure operation on the algebraic near sup-lattice $2^S {-}\{\varnothing \}$ with ${\text {im}}\!\! \downarrow \ = {\mathsf {Idl }}(S)$.

    • iii) If S has a least element, then the operation $\downarrow$ extends uniquely to a finitary closure operation on the algebraic sup-lattice $2^S$ such that $\downarrow \!\! \varnothing = \inf S$.

  • b) Let L be an algebraic near sup-complete poset and S a join semilattice.

    • i) ${\mathsf {Idl }}(S)$ is an algebraic near sup-complete poset.

    • ii) ${\textsf {K }}(L)$ is a sub join semilattice of L.

    • iii) The map $S \longrightarrow {\textsf {K }}({\mathsf {Idl }}(S))$ acting by $x \longmapsto \downarrow \!\! x$ is a poset isomorphism.

    • iv) The map ${\mathsf {Idl }}({\textsf {K }}(L)) \longrightarrow L$ acting by $I \longmapsto \sup I$ is a poset isomorphism with inverse acting by $x \longmapsto (\downarrow \!\! x) \cap {\textsf {K }}(L)$.

• 7. A multiplicative semilattice is an ordered magma M such that M is a join semilattice and $a \sup \{x,y\} = \sup \{ax,ay\}$ and $\sup \{x,y\}a = \sup \{xa , ya\}$ for all $a,x,y \in M$ [21, Section XIV.4]. Show that the following conditions are equivalent for any ordered magma M.

  • 1) M is a multiplicative semilattice with annihilator (resp., a multiplicative semilattice).

  • 2) $a (\sup X) = \sup (a X)$ and $(\sup X) a = \sup (Xa)$ for any $a \in M$ and any finite subset (resp., any finite nonempty subset) X of M.

  • 3) The map ${\textsf {K }}(2^M) \longrightarrow M$ (resp., ${\textsf {K }}(2^M){-}\{\varnothing \} \longrightarrow M$) acting by $X \longmapsto \sup X$ is a well-defined magma homomorphism.

  • 4) $\sup (X Y) = \sup (X) \sup (Y)$ for all finite subsets (resp., all finite nonempty subsets) X and Y of M.

• 8. Using Exercises 6 and 7, prove the following.

  • a) The multiplicative semilattices with annihilator (resp., multiplicative semilattices) form a category, where a morphism is a magma homomorphism $f: M \longrightarrow M'$, with M and $M'$ multiplicative semilattices with annihilator (resp., multiplicative semilattices), such that $f(\sup X) = \sup f(X)$ for all finite subsets (resp., all finite nonempty subsets) X of M.

  • b) The precoherent prequantales (resp., precoherent near prequantales) form a category, where a morphism is magma homomorphism $f: Q \longrightarrow Q'$, with Q and $Q'$ precoherent prequantales (resp., precoherent near prequantales), such that $f({\textsf {K }}(Q)) \subseteq {\textsf {K }}(Q')$.

  • c) Let M be a multiplicative semilattice.

    • i) The operation $\downarrow : X \longmapsto \downarrow \!\! X$ is a finitary nucleus on the precoherent near prequantale $2^M {-}\{\varnothing \}$, one has ${\mathsf {Idl }}(M) = \downarrow \!\! (2^M {-}\{\varnothing \})$, and ${\mathsf {Idl }}(M)$ is a precoherent near prequantale under $\downarrow$-multiplication.

    • ii) Suppose that M is with annihilator. The operation $\downarrow : X \longmapsto \downarrow \!\! X$ is a finitary nucleus on the precoherent prequantale $2^M$, one has ${\mathsf {Idl }}(M) = \downarrow \!\! (2^M)$, and ${\mathsf {Idl }}(M)$ is a precoherent prequantale under $\downarrow$-multiplication.

  • d) Let Q be a precoherent prequantale and let M be a multiplicative semilattice with annihilator.

    • i) ${\mathsf {Idl }}(M)$ is a precoherent prequantale under $\downarrow$-multiplication.

    • ii) ${\textsf {K }}(Q)$ is a multiplicative semilattice with annihilator.

    • iii) The map $M \longrightarrow {\textsf {K }}({\mathsf {Idl }}(M))$ acting by $x \longmapsto \downarrow \!\! x$ is an isomorphism of ordered magmas.

    • iv) The map ${\mathsf {Idl }}({\textsf {K }}(Q)) \longrightarrow Q$ acting by $I \longmapsto \sup I$ is an isomorphism of ordered magmas.

    • v) If $f: Q \longrightarrow Q'$ is a morphism of precoherent prequantales, then the map ${\textsf {K }}(f): {\textsf {K }}(Q) \longrightarrow {\textsf {K }}(Q')$ given by ${\textsf {K }}(f)(x) = f(x)$ for all $x \in {\textsf {K }}(Q)$ is a morphism of multiplicative semilattices with annihilator.

    • vi) If $g: M \longrightarrow M'$ is a morphism of multiplicative semilattices with annihilator, then the map ${\mathsf {Idl }}(g): {\mathsf {Idl }}(M) \longrightarrow {\mathsf {Idl }}(M')$ given by ${\mathsf {Idl }}(g)(I) = \downarrow \!\! (g(I))$ for all $I \in {\mathsf {Idl }}(M)$ is a morphism of precoherent prequantales.

    • vii) The associations ${\textsf {K }}$ and ${\mathsf {Idl }}$ are functorial and provide an equivalence of categories between the category of precoherent prequantales and the category of multiplicative semilattices with annihilator.

  • e) Generalize part (d) to show that there is an equivalence of categories between the category of precoherent near prequantales and the category of multiplicative semilattices.

Section 6.5

• 1. Prove Lemma 6.5.1.

• 2. Verify Example 6.5.6.

• 3.

  • a) Give an example of a multiplicative lattice that is not precoherent.

  • b) Give an example of a precoherent multiplicative lattice that is not coherent.

• 4. Prove Theorem 6.5.8.

• 5. Prove Theorem 6.5.9.

• 6. Prove Theorem 6.5.10.

• 7.

  • a) Let $\star$ and $\star '$ be closure operations on an algebraic bounded complete lattice S. Show that $\inf \{\star _t, \star '_t\} = (\inf \{\star ,\star '\})_t$.

  • b) Let $\star$ and $\star '$ be nuclei on a precoherent semiprequantale Q. Show that $\inf \{\star _t, \star '_t\} = (\inf \{\star ,\star '\})_t$.

Section 6.6

• 1. Fill in the missing details in the proof of Proposition 6.6.2.

• 2. Verify statement (1) of Lemma 6.6.6.

• 3. Let M be a unital ordered magma. Prove the following.

  • a) $\star -DI (M)$ is a submagma of M.

  • b) If $X \subseteq \star -DI (M)$ is nonempty, then $\sup X \in \star -DI (M)$ if $\sup X$ exists, and $\inf X \in \star -DI (M)$ if $\inf X$ exists and X is finite.

• 4. Verify Example 6.6.10.

• 5. Prove Theorem 6.6.11.

• 6. Prove Proposition 6.6.12.

• 7. Let Q be a coherent unital residuated lattice-ordered semiquantale.

  • a) Show that, for any nucleus $\star$ on Q, there is a largest stable nucleus $\overline{\star }$ on Q that is less than or equal to $\star$.

  • b) Show that, for any nucleus $\star$ on Q, there is a largest stable finitary nucleus $\star _w$ on Q that is less than or equal to $\star$.

  • c) $(**)$ If possible, find an explicit formula for $\overline{\star }$ and for $\star _w$.

• 8. $(**)$ Let Q be a precoherent prequantale. Characterize the subsets X of Q such that $\inf \{t(a): \ a \in X\}$ is finitary.

Fig. 6.1

Three full lattices of implications