residuated lattices, involutive partially ordered semigroups, involutive semirings, Płonka sums, Frobenius quantales.

1 Introduction↩︎

Idempotent semirings are algebras of the form \((A,\lor,\cdot)\) where \((A,\lor)\) is a semilattice (with order \(x\leqslant y \iff x\lor y=y\)), \((A,\cdot)\) is a semigroup, and the semigroup operation distributes over the join. They play an important role in mathematics, logic, and theoretical computer science, since they generalize distributive lattices and, if they have an identity element, they expand to Kleene algebras and residuated lattices. An involutive semiring is an idempotent semiring with two antitone operations \(\mathord{\sim}\) and \(\mathord{-}\) satisfying \(\mathord{\sim}\mathord{-}x=x=\mathord{-}\mathord{\sim}x\) and \[x\cdot y\leqslant z\iff y\cdot\mathord{\sim}z\leqslant\mathord{\sim}x \iff \mathord{-}z\cdot x\leqslant\mathord{-}y.\] These algebras are term-equivalent to identity-free involutive residuated lattices and, in the case that the lattice is complete, to Frobenius quantales (see [1] and [2]). Furthermore, algebras of binary relations are involutive semirings under the operations of union, composition, and complement-converse. The structural characterization obtained in this paper is valid for more general partially ordered structures called involutive po-semigroups (ipo-semigroups) where the semilattice \((A,\lor)\) is replaced by a poset \((A,\leqslant)\). In these structures, the product is residuated and its residuals \(\backslash\) and \(/\) are term-definable. Prominent examples of ipo-semigroups are (1-free reducts of) groups (where the poset is an antichain) and partially ordered groups (in both cases \(\mathord{\sim}x=-x=x^{-1}\)).

An ipo-semigroup has a global identity if there is an element \(1\) satisfying \(x\cdot 1 = x = 1\cdot x\), which implies the inequality \(y\leqslant(x/x)y\). An ipo-semigroup is integral when it possesses a global identity which is moreover the top element, that is, \(x\leqslant 1\). Integrality is an important property for residuated lattices, since it is equivalent to the proof-theoretical rule called weakening. Every integral ipo-semigroup is integrally closed (see [3]), namely, \(x\backslash x = 1 = x/x\), and satisfies \(x\backslash 1 = 1\). In this work we investigate a much larger class of structures, which we call locally integral ipo-semigroups. A balanced ipo-semigroup, (i.e., one satisfying \(1_x:= x\backslash x = x/x\)) is locally integral if it satisfies \(y\leqslant 1_x y\) (and in particular, \(1_x\cdot x = x\)), \(x\leqslant 1_x\), and \(x\backslash 1_x = 1_x\). For this reason, we call the elements of the form \(1_x\) local identities. An integral ipo-semigroup is a locally integral one satisfying \(1_x = 1_y\).

The main result in this paper is that every locally integral ipo-semigroup \(\mathbf{A}\) can be decomposed in a unique way into a family of integral ipo-monoids \(\{\mathbf{A}_p : p\in A^+\}\), indexed on the set \(A^+ = \{{p\in A} : \text{for every }x,\;x\leqslant px\}\) of positive¹ elements of \(\mathbf{A}\), which we call its integral components. Two locally integral ipo-semigroups can have the same integral components, but may differ in the way these components are glued together. We find in the literature similar situations in which a number of structures are glued together to form a new one: for instance, in [4] it is described how chains can be attached to an odd Sugihara monoid in order to form a commutative idempotent residuated chain, and in [5] how Boolean algebras can be glued together to form commutative idempotent involutive residuated lattices.

In our present case, we associate to every locally integral ipo-semigroup \(\mathbf{A}\) the join-semilattice \(\mathbf{A}^+ = (A^+,\cdot)\) and a family of monoid homomorphisms \(\Phi = \{\varphi_{pq}\colon {\mathbf{A}_p\to \mathbf{A}_q} : p\leqslant q \text{ in } A^+\}\) between its integral components such that the structure of \(\mathbf{A}\) can be completely recovered as an aggregate or glueing \(\int_\Phi \mathbf{A}_p\) of these integral components along \(\Phi\) in two stages: first, the semigroup part of \(\mathbf{A}\) turns out to be the Płonka sum of the family \(\Phi\), and the involutive negations can be defined componentwise. Then, we recover the order of \(\mathbf{A}\) using the product, the negations, and the local identities.

As an application of our results, we can combine certain semantics for fuzzy logics with semantics for relevance logic using, for example, the well-understood MV-algebras as building blocks of a glueing. We also exploit this decomposition in order to prove that several properties of locally integral ipo-semigroups are local, in that a locally integral ipo-semigroup satisfies them if and only if all its integral components satisfy them. One of the most significant local properties established here is local finiteness.

Previous research into the structure of doubly-idempotent semirings can be found in [6], [7]. The structure of all finite commutative idempotent involutive residuated lattices is completely described in [5] in a step-by-step decomposition. In the current paper, this is significantly generalized to all locally integral ipo-semigroups, without any restrictions regarding finiteness, commutativity, or full idempotence. Similar results were originally proved for the special case of locally integral ipo-monoids [8]. In the present form they are valid for locally integral Frobenius quantales (without a unit). We also characterize the ipo-semigroups that can be embedded into ipo-monoids. A further use of Płonka sums can be found in [9], where the structure of even and odd involutive commutative residuated chains is studied.

We set the terminology and notation in Section 2, and describe the fundamental properties of ipo-semigroups needed in the rest of the paper. In Section 3, we introduce the class of locally integral ipo-semigroups and show that every locally integral ipo-semigroup is the glueing of its integral components. Finally, in Section 4, we solve the reverse problem, that is, we provide necessary and sufficient conditions so that the glueing of a system of integral ipo-monoids is an ipo-semigroup. In the final section we consider the category of idempotent ipo-semigroups and describe its duality with a subcategory of semilattice directed systems of sets and partial maps.

2 Involutive partially ordered semigroups and semirings↩︎

An involutive partially ordered semigroup, or ipo-semigroup for short, is a structure of the form \(\mathbf{A} = (A,\leqslant,\cdot,\mathord{\sim},\mathord{-})\) such that \((A,\leqslant)\) is a partially ordered set, \((A,\cdot)\) is a semigroup with two antitone (i.e., order-reversing) unary operations \(\mathord{\sim}\) and \(\mathord{-}\) satisfying double negation (dn) \(\mathord{\sim}\mathord{-}x = x = \mathord{-}\mathord{\sim}x\) and rotation: \[x\cdot y \leqslant z \iff y\cdot\mathord{\sim}z \leqslant\mathord{\sim}x \iff \mathord{-}z\cdot x\leqslant\mathord{-}y. \] This is then a po-variety, in the sense of [10], namely, a class of partially ordered algebras that satisfy a set of equations and inequations. The unary operations \(\mathord{\sim}\) and \(\mathord{-}\), which are allowed to coincide, are called involutive negations. Indeed, we call \(\mathbf{A}\) cyclic if it satisfies \(\mathord{\sim}x = \mathord{-}x\). An element \(x\) of \(\mathbf{A}\) is central if \(x\cdot y = y \cdot x\) for any other \(y\in A\), and \(\mathbf{A}\) is commutative if all its elements are central. An element \(x\) of \(\mathbf{A}\) is idempotent if \(x\cdot x = x\), and \(\mathbf{A}\) is idempotent if all its elements are idempotent. From now on, if no confusion is likely, we will write \(xy\) instead of \(x\cdot y\).

We can readily check that, in the presence of antitonicity and double negation, rotation is equivalent to the following property of residuation: \[xy\leqslant z \iff x \leqslant\mathord{-}(y\cdot\mathord{\sim}z) \iff y \leqslant\mathord{\sim}(\mathord{-}z\cdot x). \] Thus, the multiplication of every ipo-semigroup is residuated in both arguments, with left and right residuals given by \(z/y = \mathord{-}(y\cdot\mathord{\sim}z)\) and \(x\backslash z = \mathord{\sim}(\mathord{-}z\cdot x)\), respectively. With this notation, (res) can be rewritten as follows: \[xy \leqslant z \iff x \leqslant z/y \iff y \leqslant x\backslash z.\] An immediate consequence of this observation is that multiplication preserves arbitrary existing joins and is therefore order-preserving in both arguments. Also, an ipo-semigroup is commutative if and only if it satisfies \(x\backslash y = y/x\).

Using the involutive negations and the multiplication, we can define a binary operation \(+\) on \(A\) in two different ways: \(x+y = \mathord{\sim}(\mathord{-}x\cdot\mathord{-}y)\) and \(x+y = \mathord{-}(\mathord{\sim}x\cdot \mathord{\sim}y)\). The previous lemma guarantees that both definitions agree, since \[\mathord{\sim}(\mathord{-}x\cdot \mathord{-}y) = \mathord{-}y\backslash x = y/\mathord{\sim}x = \mathord{-}(\mathord{\sim}x\cdot\mathord{\sim}y).\]

An element \(p\) of an ipo-semigroup \(\mathbf{A}\) is positive if for all \(x\in A\), \(x\leqslant px\), or equivalently, for all \(x\in A\), \(x\leqslant xp\). Indeed, suppose that for all \(x\in A\), the element \(p\) satisfies \(x\leqslant px\). In particular, \(\mathord{\sim}(xp) \leqslant p\cdot\mathord{\sim}(xp)\), whence we obtain that \[x\leqslant(xp)/p = \mathord{-}(p\cdot\mathord{\sim}(xp)) \leqslant\mathord{-}\mathord{\sim}(xp) = xp,\] by residuation, antitonicity, and double negation. The other implication is analogous. Notice that this also implies both that \(p\backslash x \leqslant x\) and \(x/p \leqslant x\). We denote by \(A^+\) the set of all positive elements of \(\mathbf{A}\), which one can readily prove is closed upwards, that is, if \(p\in A^+\) and \(p\leqslant q\), then \(q\in A^+\).

An element 1 of an ipo-semigroup \(\mathbf{A}\) is a global identity if \(1\cdot x = x\), for all \(x\in A\). In particular, a global identity is always positive, and therefore \(x\leqslant x\cdot 1\). At the same time, \(1\cdot\mathord{\sim}x = \mathord{\sim}x\) implies that \(1\leqslant\mathord{\sim}x/\mathord{\sim}x = x\backslash x\) and hence \(x\cdot 1 \leqslant x\), by residuation and contraposition. Thus, global identities also satisfy \(x\cdot 1 = x\), for all \(x\in A\), and therefore an ipo-semigroup cannot have more than one global identity. In that case, the structure \(\widehat{\mathbf{A}} = (A,\leqslant,\cdot, 1, \mathord{\sim}, \mathord{-})\) is an ipo-monoid as defined in [8] and the set \(A^+\) of positive elements of \(\mathbf{A}\) coincides with the positive cone \(\{p\in A : 1\leqslant p\}\) of \(\widehat{\mathbf{A}}\). Indeed, an ipo-monoid is a structure \((A,\leqslant, \cdot, 1,\mathord{\sim}, \mathord{-})\) consisting of a poset \((A,\leqslant)\), a monoid \((A,\cdot, 1)\), and two unary operations \(\mathord{\sim}\) and \(\mathord{-}\) satisfying the following condition: \[x\cdot\mathord{\sim}y\leqslant 0 \iff x\leqslant y \iff \mathord{-}y\cdot x \leqslant 0, \] where \(0=\mathord{-}1\). Then, if \(\mathbf{A}\) is an ipo-semigroup with a global identity \(1\), we have that \((A,\leqslant)\) is a poset and \((A,\cdot, 1)\) is a monoid. Also, \(1\cdot\mathord{-}1\leqslant\mathord{-}1\) implies \(\mathord{-}1 = \mathord{-}1\cdot 1 = \mathord{-}1\cdot\mathord{\sim}\mathord{-}1 \leqslant\mathord{\sim}1\), by (dn) and (rot). Analogously, \(\mathord{\sim}1\leqslant\mathord{-}1\), and therefore \(\mathord{-}1 = \mathord{\sim}1\). Hence, rotation and the fact that \(1\cdot x = x = x\cdot 1\) imply (ineg). On the other hand, the 1-free reduct of any ipo-monoid is an ipo-semigroup.

We say that \(\mathbf{A}\) is integral if it possesses a global identity \(1\) which is also the top element of the order of \(\mathbf{A}\). The confluence of these two properties on \(1\) imposes a number of restrictions on \(\mathbf{A}\). For instance, since \(1\cdot x = x = x\cdot 1\), we obtain by residuation that \(1 \leqslant x/x\) and \(1\leqslant x\backslash x\), and since \(1\) is the top element, we deduce that \(x\backslash x = 1 = x/x\). Also, since \(1\cdot x \leqslant 1\) and \(x\cdot 1\leqslant 1\), we also have that \(1 \leqslant 1/x\) and \(1 \leqslant x\backslash 1\), and therefore \(x\backslash 1 = 1 = 1/x\). The next lemma characterizes the integral ipo-semigroups as the ones satisfying two particular inequations. As a consequence, we obtain that the class of integral ipo-semigroups is a po-subvariety of the po-variety of ipo-semigroups, in the sense of [10].

The following lemma about ipo-semigroups will be used numerous times in the following sections.

We end this section by underlining a special class of ipo-semigroups, namely those which have a lattice order. These can be then presented as algebraic structures \((A,\land,\lor,\cdot,\mathord{\sim},\mathord{-})\) called i\(\ell\)-semigroups or involutive semirings² and integral i\(\ell\)-semigroups or integral involutive semirings, if they are integral as ipo-semigroups. Notice that, since the inequality \(xy \leqslant y\) can be expressed as \(xy \lor y = y\) in the language of involutive semirings, the integral involutive semirings form a subvariety of the variety of involutive semirings.

Note that the variety of unital i\(\ell\)-semirings (without the assumption of integrality) coincides with the variety of involutive residuated lattices, and unital Frobenius quantales are the same as complete unital i\(\ell\)-semirings.

3 Locally integral ipo-semigroups and involutive semirings↩︎

An ipo-semigroup \(\mathbf{A}\) has local identities if for all \(x\in A\), \(x\backslash x = x/x\), in which case we use \(1_x\) to denote this element, and \(1_x\cdot x = x\). Notice that, if \(\mathbf{A}\) has local identities, by Lemma 3, we also have that for all \(x\in A\), \(\mathord{-}x\cdot x = x\cdot\mathord{\sim}x\) and we use \(0_x\) to denote this element. If \(\mathbf{A}\) possesses a global identity \(1\), then it is the smallest of the local identities: this is because \(1\backslash 1 = 1 = 1/1\) and then \(1_1 = 1\), and \(1\cdot x = x\) implies that \(1\leqslant x/x = 1_x\), for all \(x\in A\). Hence, if \(\mathbf{A}\) has a global identity, all local identities are positive. The following are other fundamental properties of local identities.

Inspired by the notion of an integral ipo-semigroup and its elementary properties, we define an ipo-semigroup \(\mathbf{A}\) to be locally integral if it has local identities (\(1_x = x\backslash x = x/x\) and \(1_x\cdot x = x\)) satisfying, moreover, the following conditions for all \(x,y\in A\):

1. the local identities are positive, that is, \(y\leqslant 1_xy\),

2. the elements are locally bounded by their identities, that is, \(x\leqslant 1_x\),

3. and \(x\backslash 1_x = 1_x\).

As we have seen before, in an integral ipo-semigroup with global identity \(1\) we have that \(x\backslash x = 1 = x/x\), hence \(1_x = 1\) and \(1_x\cdot x = 1\cdot x = x\). Thus, all local identities coincide with 1, which is positive and also satisfies \(x\leqslant 1\) and \(x\backslash 1 = 1\). That is, every integral ipo-semigroup is locally integral. The following proposition characterizes local integrality for ipo-semigroups and shows that if a locally integral ipo-semigroup \(\mathbf{A}\) has a global identity, then the corresponding \(\widehat{\mathbf{A}}\) is a locally integral ipo-monoid as defined in [8], and the 1-free reducts of the locally integral ipo-monoids are locally integral ipo-semigroups.

The main goal of this section is a decomposition theorem stating that every locally integral ipo-semigroup (involutive semiring, respectively) can be decomposed in a very particular way into integral involutive ipo-semigroups (involutive semirings, respectively). But, before addressing this decomposition, we need a better description of the positive elements of a locally integral ipo-semigroup \(\mathbf{A}\). Recall that \(A^+ = \{p\in A : \text{for all }x, x\leqslant px\} = \{p\in A : \text{for all }x, x\leqslant xp\}\) and that by definition \(1_x\in A^+\), for all \(x\in A\). We also define the set \(A^\sim = \{\mathord{\sim}p : p\in A^+\}\), which is closed downwards, since \(A^+\) is closed upwards and \(\mathord{\sim}\) is antitone.

Given a locally integral ipo-semigroup \(\mathbf{A}\), the sets \(A^+\) and \(A^\sim\) are obviously partially ordered by the order of \(\mathbf{A}\). It turns out that \((A^+,\leqslant)\) and \((A^\sim,\leqslant)\) are semilattices dual to each other, and their properties are very consequential. For instance, if \((A^+,\leqslant)\) has a minimum element \(p\) then it is the global identity of \(\mathbf{A}\), because for every \(a\in A\) we have that \(p\leqslant 1_a = a/a\), and therefore \(pa\leqslant a\); but since \(p\) is positive, we also have that \(a\leqslant pa\), showing that \(pa=a\). The next proposition describes these two posets in more detail.

In order to obtain the aforementioned decomposition theorem, we start by noticing that the equivalence relation \(x \equiv y\) if and only if \(1_x = 1_y\) partitions every locally integral ipo-semigroup into its equivalence classes \(A_x = \{y\in A : 1_x = 1_y\}\) and, obviously, \(x\in A_x\). The next lemma offers a very useful description of \(A_x\) in which we make use of the interval notation \([0_x,1_x] = \{y\in A : 0_x \leqslant y \leqslant 1_x\}\). Recall that by virtue of Lemma 4 and Proposition [prop:char:locally:integral], both \(1_x\) and \(0_x\) are idempotent. This implies that every interval \([0_x,1_x]\) is closed under products. Indeed, if \(y,z\in [0_x,1_x]\) then \(0_x = 0_x0_x \leqslant yz \leqslant 1_x1_x = 1_x\), by monotonicity, and hence \(yz\in [0_x,1_x]\).

Next, we will use the description of \(A_x\) of the previous lemma in order to show that the sets \(A_x\) are closed under several operations of \(\mathbf{A}\).

Our next goal is to find a canonical representative for each equivalence class \(A_x\).

We can now show that the relation and operations of a locally integral ipo-semigroup furnish each equivalence class \(A_p\) with the structure of an integral ipo-monoid \(\mathbf{A}_p\). We call every \(\mathbf{A}_p\) an integral component of \(\mathbf{A}\).

What we have proven so far is that every locally ipo-semigroup \(\mathbf{A}\) can be broken into a disjoint union of a family of ipo-monoids \(\{\mathbf{A}_p : p\in A^+\}\). But, obviously, \(\mathbf{A}\) is not the mere “union” of these ipo-monoids: the elements from different components can be multiplied and compared with each other. This is the content of the following two lemmas.

All these results point toward the idea that locally integral ipo-semigroups are built up from integral ipo-monoids, or at least their semigroup reducts are, by means of a Płonka sum. This construction was first introduced and studied in [11]–[13]; for more recent expositions see [14] and [15]. Let \((I, \lor)\) be a join-semilattice. A family of homomorphisms \(\{\varphi_{ij}\colon \mathbf{A}_i\to \mathbf{A}_j : i\leqslant j \in I\}\) between algebras of the same type is said to be compatible if for every \(i\in I\), \(\varphi_{ii}\) is the identity on \(\mathbf{A}_i\), and that if \(i \leqslant j \leqslant k\) then \(\varphi_{jk}\circ\varphi_{ij} = \varphi_{ik}\). Given such a compatible family of homomorphisms, its Płonka sum is an algebra \(\mathbf{S}\) defined on the disjoint union of their universes \(S = \biguplus_{i\in I} A_i\). For every nonconstant \(n\)-ary operation symbol \(\sigma\) and elements \(a_1\in A_{i_1}\), …, \(a_n\in A_{i_n}\), \(\sigma^{\mathbf{S}}(a_1,\dots,a_n) = \sigma^{\mathbf{A}_j}(\varphi_{i_1j}(a_1),\dots,\varphi_{i_nj}(a_n))\), where \(j = i_1\lor \dots\lor i_n\). One can readily prove that the Płonka sum of a compatible family of homomorphisms is well-defined and it satisfies all regular equations that hold in all the algebras of the family which do not involve constants. Recall that a regular equation is an equation in which the variables that appear on the left-hand side are the same as the variables that appear on the right-hand side.

Given a locally integral ipo-semigroup \(\mathbf{A}\), consider the map \(\varphi_{pq}\colon A_p \to A_q\) given by \(\varphi_{pq}(x) = qx\), for every pair of positive elements \(p \leqslant q\). We will show that \(\Phi = \{\varphi_{pq} : p\leqslant q\}\) is a compatible family of monoid homomorphisms between the family of algebras \(\{\mathbf{A}_p : p\in A^+\}\), and that the semigroup reduct of \(\mathbf{A}\) can be reconstructed as the Płonka sum of \(\Phi\).³

As we will show next, the semigroup reduct of a locally integral ipo-semigroup is the Płonka sum of (the semigroup reducts of) the family above. Moreover, although this is not the case for the rest of the structure, we can still recover it from its integral components by virtue of Lemma 10. Also, we saw in Proposition [prop:integral:components] that some properties of \(\mathbf{A}\) are inherited by each of its integral components. Sometimes the converse is also true. We call a property of ipo-semigroups local whenever an ipo-semigroup has it if and only if all its integral components have it.

The previous theorem covers the structural decomposition results in [5] (see Theorem 4 and Corollary 4 below). In that paper it is also shown that the variety of commutative idempotent involutive residuated lattices fails to be locally finite. Without the lattice operations, however, we have the following result.

4 Glueing Constructions↩︎

The last theorem of the previous section shows how every ipo-semigroup is an aggregate of its integral components. Our next question is, what are the conditions that a family of integral ipo-monoids and a compatible family of monoid homomorphisms between them should satisfy so that the construction of Theorem 1 is a (locally integral) ipo-semigroup?

To make this question precise, suppose that \(\mathbf{D} = (D,\lor)\) is a join-semilattice, \(\mathcal{A} = \{\mathbf{A}_p : p\in D\}\) is family of integral ipo-monoids, and \(\Phi = \{\varphi_{pq}\colon \mathbf{A}_p\to \mathbf{A}_q : p \leqslant^{\mathbf{D}} q\}\) is a compatible family of monoid homomorphisms. We call \((\mathbf{D}, \mathcal{A}, \Phi)\) a semilattice directed system of integral ipo-monoids.⁴ Letting \(\mathbf{A}_p = (A_p, \leqslant_p, \cdot_p, 1_p, \mathord{\sim}_p, \mathord{-}_p)\), for all \(p\) in \(D\), we define the structure \[{\textstyle\int_\Phi \mathbf{A}_p} = \big(\textstyle\biguplus\nolimits_D A_p, \leqslant, \cdot, \mathord{\sim}, \mathord{-}\,\big),\] where \(\big(\biguplus_D A_p, \cdot \big)\) is the Płonka sum of the family \(\Phi\), and therefore a semigroup, and for all \(p,q\in D\), \(a\in A_p\), and \(b\in A_q\), we define \(\mathord{\sim}a = \mathord{\sim}_p a\) and \(\mathord{-}a = \mathord{-}_p a\), and \[a\leqslant b \iff a\cdot \mathord{\sim}b = 0_{p\lor q}.\] We call this structure \(\int_\Phi \mathbf{A}_p\) the glueing of \(\mathcal{A}\) along the family \(\Phi\). With these definitions, one can see that given a locally integral ipo-semigroup \(\mathbf{A}\), its semilattice of positive elements \(\mathbf{A}^+\), its family \(\mathcal{A}\) of integral components, and its associated family \(\Phi=\{\varphi_{pq}\colon \mathbf{A}_p\to \mathbf{A}_q\}\) of monoid homomorphisms determined by \(\varphi_{pq}(x) = qx\), we have that \((\mathbf{A}^+, \mathcal{A}, \Phi)\) is a semilattice directed system of integral ipo-monoids. We can restate Theorem 1 as saying that every locally integral ipo-semigroup \(\mathbf{A}\) is the glueing \(\int_\Phi \mathbf{A}_p\) of its integral components along the family of homomorphisms \(\Phi\).⁵

The last proposition can be interpreted as saying that the glueing \(\int_\Phi \mathbf{A}_p\) is indeed an “aggregate” of the integral ipo-monoids \(\mathbf{A}_p\), although not necessarily an ipo-semigroup itself, since the relation \(\leqslant\) could fail to be antisymmetric and transitive. For instance, in the example of Figure 5, \(0_r \leqslant 0_p \leqslant 1_p \leqslant 1_q\), but \(0_r \nleqslant 1_q\). The natural question is then: given a semilattice directed system \((\mathbf{D}, \mathcal{A}, \Phi)\) of integral ipo-monoids, what are the conditions that \(\Phi\) must satisfy in order to ensure that \(\int_\Phi \mathbf{A}_p\) is an ipo-semigroup?

To answer this, our first step will be to examine the sets \(G^+ = \{a \in \biguplus A_p : x \leqslant a\cdot x, \text{for all } x\}\) and \(G^\sim = \{\mathord{\sim}a : a \in G^+\}\) associated to any given glueing \(\mathbf{G} = \int_\Phi\mathbf{A}_p\). As a corollary to Proposition [prop:glueing:is:an:aggregate], we will show that the elements of \(G^+\) are the elements of the form \(1_p\), and the elements of \(G^\sim\) are the ones of the form \(0_p\), for some \(p\in D\).

Reflecting on Proposition [prop:A43:join:semilattice], we would like to show that the relation \(\leqslant\) endows \(G^+\) with the structure of a join-semilattice isomorphic to \(\mathbf{D}\), and \(G^\sim\) with the structure of a meet-semilattice dually isomorphic to \(\mathbf{D}\). In general, this will not be true. For this to hold, it will be necessary to assume an extra property of \(\Phi\). We will prove first that this property is valid for the family of monoid homomorphisms associated to a locally integral ipo-semigroup.

This suggests the following condition for \(\Phi\), which we call zero avoidance: \[\label{eq:condition:on:zeros} \text{for all } p \leqslant^{\mathbf{D}} q,\quad \varphi_{pq}(0_p) = 0_q \iff p = q. \tag{1}\]

Next, recall that by virtue of Lemma 10, in any locally integral ipo-semigroup \(\mathbf{A}\), for all positive elements \(p\) and \(q\) and \(x\in A_p\) and \(y\in A_q\), we have that \(x\leqslant y \iff x\cdot\mathord{\sim}y = 0_{pq} \iff \mathord{-}y\cdot x = 0_{pq}\). Therefore, for a glueing \(\int_\Phi \mathbf{A}_p\) to be a locally integral ipo-semigroup, the family \(\Phi\) has to satisfy the corresponding condition: \[\label{eq:cond:equivalent:to:Phi:satisfies:bal} \text{for all } p,q\in D, a\in A_p, b\in A_q,\quad a\cdot\mathord{\sim}b = 0_{p\lor q} \iff \mathord{-}b\cdot a = 0_{p\lor q}.\tag{2}\]

As we will see in the next lemma, this is equivalent to the requirement that all the morphisms in \(\Phi\) satisfy the following balance property: \[\label{eq:Phi:satisfies:bal} \text{for every } a\in A_p,\quad \mathord{\sim}\varphi_{pq}(\mathord{-}a) = \mathord{-}\varphi_{pq}(\mathord{\sim}a).\tag{3}\]

We say that \(\Phi\) is balanced if all the morphisms in \(\Phi\) are balanced. One can readily check that the commutativity of \(\int_\Phi \mathbf{A}_p\) implies that \(\Phi\) is balanced. We will prove next that when \(\Phi\) is balanced, the operations \(\mathord{\sim}\) and \(\mathord{-}\) are involutive with respect to the relation \(\leqslant\) and the glueing satisfies rotation.

The semilattice directed system of Figure 5 avoids zeros and is balanced —since its components are commutative, and therefore cyclic—, but the glueing is not an ipo-semigroup because the relation \(\leqslant\) is not transitive. Hence, we also need to impose the following condition on the family \(\Phi\) for \(\int_\Phi \mathbf{A}_p\) to be an ipo-semigroup: \[\label{eq:transitivity:leG} \text{for all } a,b,c\in \textstyle\biguplus A_p,\quad \text{if } a \leqslant b \text{ and } b \leqslant c, \text{ then } a \leqslant c. \tag{4}\] Our next result characterizes the condition 4 in simpler terms.

We are now in the position to prove our main result.

The last theorem is the object correspondence of a categorical equivalence. First, for any arbitrary category \(\mathsf C\), we can define the category \(\mathsf C^\mathrm{SD}\) of semilattice directed systems on \(\mathsf C\) as the category whose objects are functors \(\Phi\colon\mathsf D\to \mathsf C\), where \(\mathsf D\) is a skeletal and thin category with binary coproducts (i.e., a join-semilattice). A morphism \(\Phi\to\Psi\) in \(\mathsf C^\mathrm{SD}\) is a pair \(\langle\tau,\eta\rangle\), where \({\tau\colon\mathsf D_\Phi\to\mathsf D_\Psi}\) is a coproduct-preserving functor and \(\eta\colon \Phi\xrightarrow{.}\Psi\tau\) is a natural transformation.

Now, let \(\LIIPOSg\) be the category whose objects are the locally integral ipo-semigroups and morphisms are the monotone algebraic homomorphisms. Let \(\IIPOMn\) be the category of integral ipo-monoids and monoid homomorphisms, and let \(\SDIIPOMn\) be the full subcategory of \(\IIPOMn^\mathrm{SD}\) whose objects are the functors \(\Phi\colon\mathsf{D}\to \IIPOMn\) satisfying 1 , 3 , (mon), and (lax). ⁶

Every morphism \(\eta\colon\mathbf{A}\to\mathbf{B}\) induces a semilattice homomorphism \(\tau\colon \mathbf{A}^+\to\mathbf{B}^+\) and monoid homomorphisms \(\eta_p\colon\mathbf{A}_p\to \mathbf{B}_{\tau(p)}\) —which are the corresponding restrictions of \(\eta\)— such that \[\text{for all p\leqslant q in A^+,}\quad \eta_q\varphi_{pq}^\mathbf{A} = \varphi_{\tau(p)\tau(q)}^\mathbf{B}\eta_p.\] Conversely, any semilattice homomorphism \(\tau\colon\mathbf{A}^+\to\mathbf{B}^+\) and family of monoid homomorphisms \(\eta_p\colon\mathbf{A}_p\to\mathbf{B}_{\tau(p)}\) satisfying (nat) induce a morphism \(\eta\colon\mathbf{A}\to\mathbf{B}\). That is, every morphism \(\eta\colon\mathbf{A}\to \mathbf{B}\) in \(\LIIPOSg\) corresponds to a morphism \(\langle\tau,\eta\rangle\colon\Phi^\mathbf{A}\to\Phi^\mathbf{B}\) in \(\SDIIPOMn\). For the other direction, every object \(\Phi\) of \(\SDIIPOMn\) gives rise to an object \(\int\Phi\) in \(\LIIPOSg\), and every morphism \({\langle\tau,\eta\rangle\colon\Phi\to\Psi}\) induces a morphism \(\eta\colon\int\Phi\to\int\Psi\) in \(\LIIPOSg\). This correspondence is an equivalence of categories between \(\LIIPOSg\) and \(\SDIIPOMn\).

A morphism in \(\LIIPOSg\) is an embedding if it is order-reflecting, and therefore injective. \(\mathbf{A}\) is a substructure of \(\mathbf{B}\) if and only if the inclusion map \(A\to B\) is an embedding, that is, if \(\mathbf{A}^+\) is a subsemilattice of \(\mathbf{B}^+\), for every \(p\in A^+\), \(\mathbf{A}_p\) is a substructure of \(\mathbf{B}_p\), and for all \(p\leqslant q\) in \(A^+\), the associated monoid homomorphisms \(\varphi_{pq}^\mathbf{A}\colon\mathbf{A}_p\to\mathbf{A}_q\) is the restriction of the corresponding \(\varphi_{pq}^\mathbf{B}\colon\mathbf{B}_p\to \mathbf{B}_q\). Also, \(\mathbf{A}\) is isomorphic to \(\mathbf{B}\) if and only if there is a semilattice isomorphism \(\tau\colon\mathbf{A}^+\to \mathbf{B}^+\) and isomorphisms \(\eta_p\colon\mathbf{A}_p\to \mathbf{B}_{\tau(p)}\) for all \(p\in A^+\) satisfying (nat).

We have already characterized which locally integral ipo-semigroups are the \(1\)-free reducts of the locally integral ipo-monoids, namely those that have a global identity, or equivalently, those that have a smallest positive element. We end this section by responding to a very natural question: which locally integral ipo-semigroups are the subreducts of locally integral ipo-monoids? Notice that, by the discussion above, checking if a locally integral ipo-semigroup is a substructure of or isomorphic to another one can be done componentwise.

5 Idempotent locally integral ipo-semigroups↩︎

In this section we specialize to locally integral ipo-semigroups in which all elements are idempotent, i.e., the identity \(x^2=x\) holds. The following two results describe this class of structures.

Since a commutative and idempotent semigroup is a semilattice, we will refer to these structures as locally integral ipo-semilattices, or i\(\ell\)-semilattices in case they are lattice-ordered. This semilattice order, called the multiplicative order, is denoted by \(\sqsubseteq\) and it is defined by \(x\sqsubseteq y\iff xy=x\).

The smallest locally integral i\(\ell\)-semilattice that doesn’t have a global unit is the \(4\)-element algebra described in Example 2. And the smallest locally integral ipo-semilattice that is not lattice-ordered has \(8\) elements. Their posets and multiplicative orders are shown in Figure 8.

The number of locally integral ipo-semilattices with \(n\leqslant 16\) elements (up to isomorphism) have been calculated with Mace4 [16] and are given in Table 2. Apart from the 16-element Boolean algebra, all these po-algebras can be constructed from glueings of Boolean algebras of cardinality \(\leqslant 8\).

To end this section, we will describe a duality for a certain class of locally integral ipo-semilattices including all finite ones. This dual representation gives a much more compact way of drawing finite ipo-semilattices (see Figures 9 and 10). We first recall that generalized Boolean algebras form the variety GBA generated by the 2-element residuated lattice \(\mathbf{2}_r\). An equational basis for this variety is given by the identities for residuated lattices together with \(xy=x\land y\) and \((y\backslash x)/ y=x\lor y\) (see [1]). Actually, GBAs can be characterized as the residuated lattices for which every principal filter is a Boolean lattice (see [1]).

Finite members of GBA are reducts of Boolean algebras without complementation or a constant designating the bottom element. Homomorphisms between GBAs are residuated lattice homomorphisms, hence they preserve the top element \(1\), join, meet, and residuals, but not necessarily complementation or \(0\). A generalized Boolean algebra is complete if all joins (and therefore all meets) exist, and it is atomic if every non-bottom element is above some atom, or equivalently, if every element is the join of all the atoms below it.

Consider the category \(\CAGBA\) whose objects are the complete and atomic generalized Boolean algebras and whose morphisms are the sup-preserving homomorphisms of generalized Boolean algebras. Consider also the category \(\Par\) of sets and partial maps. The well-known Tarski duality between the category of complete and atomic Boolean algebras and the category of sets and maps can be extended to a duality between \(\CAGBA\) and \(\Par\) in the following way.

Consider the functor \(\Psi\colon\CAGBA\to\Par^\mathrm{op}\) defined as follows: for every \(\mathbf{B}\) in \(\CAGBA\), \(\Psi(\mathbf{B})=\mathop{\mathrm{At}}(\mathbf{B})\), the set of atoms of \(\mathbf{B}\), and for every morphism \(\theta\colon\mathbf{B}\to \mathbf{B}'\), \(\Psi(\theta)\colon\mathop{\mathrm{At}}(\mathbf{B}')\nrightarrow\mathop{\mathrm{At}}(\mathbf{B})\) is the partial map defined on \(\{b\in\mathop{\mathrm{At}}(\mathbf{B}') : b\nleqslant\theta(0_\mathbf{B})\}\) as \(\Psi(\theta)(b)\) being the only \(a\in\mathop{\mathrm{At}}(\mathbf{B})\) such that \(b\leqslant\theta(a)\), and remaining undefined otherwise. Notice that this is a well-defined partial map, because if \(b\in\mathop{\mathrm{At}}(\mathbf{B}')\) and \(a,a'\in\mathop{\mathrm{At}}(\mathbf{B})\) are such that \(b\leqslant\theta(a)\) and \(b\leqslant\theta(a')\), then \(b\leqslant\theta(a)\land \theta(a') =\theta(a\land a') =\theta(0_\mathbf{B})\). In the other direction, consider the functor \(\mathcal{P}\colon\Par^\mathrm{op}\to\CAGBA\) defined for every \(X\) as \(\mathcal{P}(X)\), the powerset generalized Boolean algebra, and for every partial map \(f:X\nrightarrow Y\), \(\mathcal{P}(f)\colon\mathcal{P}(Y)\to\mathcal{P}(X)\) is the partial inverse image defined as \(\mathcal{P}(f)(a) = U_f\cup\{x\in X : f(x)\in a\}\), where \(U_f = \{x\in X : f(x)\text{ is undefined}\}\). These two functors establish the aforementioned categorical equivalence between \(\CAGBA\) and \(\Par^\mathrm{op}\).

Now, a complete and atomic generalized Boolean algebra can be understood as an integral ipo-monoid, and every sup-preserving homomorphism between complete and atomic generalized Boolean algebras is in particular a monoid homomorphism. Thus, we can view \(\CAGBA\) as a subcategory of \(\IIPOMn\). We would like to use the duality between \(\CAGBA\) and \(\Par\) and the equivalence of categories described in the previous section in order to give a duality between the locally integral ipo-semilattices whose components are complete and atomic and a certain category of semilattice directed systems of sets and partial maps.

The functor categories \(\CAGBA^\mathrm{SD}\) and \((\Par^\mathrm{op})^\mathrm{SD}\) are equivalent, since \(\CAGBA\) and \(\Par^\mathrm{op}\) are equivalent. That said, not every semilattice directed system of complete and atomic generalized Boolean algebras will give rise via the glueing construction to a locally integral ipo-semigroup, but only those satisfying 1 , 3 , (mon), and (lax). Notice that, by assumption, all the considered homomorphisms are sup-preserving, and therefore monotone. Also, since these structures are commutative, condition 3 is trivially satisfied. Consider then the full subcategory \(\SDCAGBA\) of \(\CAGBA^\mathrm{SD}\) of those functors satisfying 1 and (lax). Using the equivalence of the previous section, this category \(\SDCAGBA\) is equivalent to the category of locally integral ipo-semilattices which are locally complete and atomic. And by the duality just described, \(\SDCAGBA\) is equivalent to a full subcategory \(\SDPar\) of \((\Par^\mathrm{op})^\mathrm{SD}\).

We note that conditions on semilattice directed systems of \(\CAGBA\) translate into conditions on the semilattice directed systems of partial maps. For instance, the condition 1 corresponds to the following condition on a functor \(F\colon\mathsf D\to\Par^\mathrm{op}\) in \(\SDPar\): for all \(p\leqslant q\) in \(\mathsf D\), \[f_{pq}\colon F(q)\nrightarrow F(p) \text{ is total}\quad\iff\quad p=q.\]

Figures 9 and 10 contain several examples of locally integral ipo-semilattices. If \(\mathbf{A}\) is finite, then in particular \(\mathbf{A}^+\) and \(\mathbf{A}_p\) are finite, for all \(p\in A^+\). Hence, its dual can be displayed as the Hasse diagram of \((\mathbf{A}^+)^\mathrm{op}\) where every node \(p\) of \(\mathbf{A}^+\) is replaced by the set \(\mathop{\mathrm{At}}(\mathbf{A}_p)\). If \(q\) is a cover of \(p\), the partial map \(f_{pq}\) is given by edges from elements of \(\mathop{\mathrm{At}}(\mathbf{A}_q)\) to elements of \(\mathop{\mathrm{At}}(\mathbf{A}_p)\). The empty set is labeled by \(\emptyset\) and the empty partial map is indicated by a dotted line between the sets. This graphical representation and the fact that systems of partial maps are logarithmically smaller than the corresponding finite ipo-semilattices make them a useful tool for investigating finite locally integral ipo-semilattices.

References↩︎