The Structure of Free Domain Semirings

P. Jipsen, G. Struth

Chapman U Sheffield

May 23, 2008

P. Jipsen, G. Struth (Chapman U Sheffield) Free Domain Semirings May 23, 2008 1 / 38 Outline

Introduction

Domain Semirings

Free domain semiring

Representation by antichains of d-sequences

Representation by binary relations

Domain complement

Domain semirings with operators

P. Jipsen, G. Struth (Chapman U Sheffield) Free Domain Semirings May 23, 2008 2 / 38 Introduction

A semiring is of the form (A, +, 0, ·, 1) such that

(A, +, 0) is a commutative monoid (A, ·, 1) is a monoid · distributes over all finite joins from the left and right i.e. x(y + z)= xy + xz, (x + y)z = xz + yz and x0 = 0x = 0

A semiring is idempotent if x + x = x

IS is the variety of idempotent semirings Lemma An idempotent semiring is a (join-)semilattice with 0 as bottom element, with x ≤ y given by x + y = y (since + is assoc, commu and idempotent) and x ≤ y =⇒ wxz ≤ wyz (since w(x + y)z = wxz + wyz)

P. Jipsen, G. Struth (Chapman U Sheffield) Free Domain Semirings May 23, 2008 3 / 38 Examples

Examples of semirings are:

Rings (N, +, 0, ·, 1) ...

Examples of idempotent semirings are:

Reducts of relation algebras (A, +, 0, ; , 1) Reducts of Kleene algebras (A, +, 0, ·, 1) Reducts of residuated lattices (A, ∨, ⊥, ·, 1) (R ∪ {−∞}, max, −∞, +, 0) Bounded distributive lattices (A, ∨, 0, ∧, 1) ...

P. Jipsen, G. Struth (Chapman U Sheffield) Free Domain Semirings May 23, 2008 4 / 38 Free monoids and semirings

Let X be a set of variables (or generators)

∗ n The free monoid over X is X = [ X with 1 = empty sequence and · as n∈N concatenation

By distributivity, every term t in the signature of semirings can be written as a finite join of terms of the free monoid X ∗

Example: x(y + xz)(x +1) = xyx + xxzx + xy + xxz

⇒ the free idempotent semiring over X , denoted by FIS(X ), is isomorphic ∗ to the set Pfin(X ) of all finite subsets of words over X

Here U + V = U ∪ V and U · V = {uv : u ∈ U, v ∈ V }

P. Jipsen, G. Struth (Chapman U Sheffield) Free Domain Semirings May 23, 2008 5 / 38 Decidability

⇒ the equational theory of idempotent semirings is decidable:

Given terms s, t, use distributivity to write terms in normal form

However, the quasiequational theory (= strict universal Horn theory) is undecidable because:

The word problem for semigroups is undecidable (Post) Every semiring is a semigroup under the operation “·”

Every semigroup S is a “·”-subreduct of its powerset semiring P(Se ) (where Se the monoid extension of S) ⇒ the class of “·”-subreducts of semirings is the class of all semigroups

P. Jipsen, G. Struth (Chapman U Sheffield) Free Domain Semirings May 23, 2008 6 / 38 Domain monoids

A domain monoid is an algebra (M, ·, 1, d) such that

(M, ·, 1) is a monoid and d : M → M is a function that satisfies

(D1) d(x)x = x (D2) d(xd(y)) = d(xy) (D3) d(d(x)y)= d(x)d(y) (D4) d(x)d(y)= d(y)d(x)

The varieties of domain monoids is denoted by DM Lemma d(1) = 1 [take x = 1 in (D1)] d(d(x)) = d(x) [take x = 1 in (D2)] d(x)d(x)= d(x) [take y = x in (D3)]

⇒ d(M)= {d(x): x ∈ M} is a meet semilattice with 1 = top element P. Jipsen, G. Struth (Chapman U Sheffield) Free Domain Semirings May 23, 2008 7 / 38 Domain semirings

A domain semiring is an algebra (A, +, 0, ·, 1, d) such that

(A, +, 0, ·, 1) is a semiring

(A, ·, 1, d) is a domain monoid and

the following additional axioms hold [Desharnais, Struth 2008]

d(x + y)= d(x)+ d(y), d(0) = 0 and d(x)+1=1

⇒ xd(x)+ x = x ⇒ x + x = x

⇒ Every domain semiring is an idempotent semiring

The varieties of domain semirings is denoted by DS

P. Jipsen, G. Struth (Chapman U Sheffield) Free Domain Semirings May 23, 2008 8 / 38 Examples of domain semirings

Examples of domain semirings are e.g.

, reducts of relation algebras with d(x) = (x;x`) ∧ 1 reducts of Kleene algebras with domain

Models of domain semirings in CS:

Idempotent semirings formed by sets of traces of a program (which are alternating sequences of state and action symbols) with domain defined by starting states of traces

Idempotent semirings formed by sets of paths in a graph with domain defined by sets of starting states

Applications of domain semirings and Kleene algebras with domain have been studied intensively

P. Jipsen, G. Struth (Chapman U Sheffield) Free Domain Semirings May 23, 2008 9 / 38 Applications of domain semirings

The domain operation models enabledness conditions for actions in programs and transition systems

The domain operation can easily be extended into a modal diamond operator that acts on the underlying algebra of domain elements [M¨oller, Struth 2006]

Links the algebraic approach with more traditional logics of programs such as dynamic, temporal and Hoare logics

Some standard semantics of programs, including the weakest precondition and weakest liberal precondition semantics, can be modeled in this setting

Applications can be found in RelMiCS conference proceedings

P. Jipsen, G. Struth (Chapman U Sheffield) Free Domain Semirings May 23, 2008 10 / 38 Domain semirings

Domain semirings were originally introduced in a two-sorted setting

The domain operation maps arbitrary semiring elements to a special Boolean subalgebra [Desharnais, M¨oller, Struth 2006]

Arbitrary semiring elements model actions of a program or transition system

The elements of the Boolean subalgebra model the states of that system

Here we use the simpler and more general one-sorted approach of [Desharnais, Struth 2008]

P. Jipsen, G. Struth (Chapman U Sheffield) Free Domain Semirings May 23, 2008 11 / 38 Studying free domain semirings

The free domain semiring is interesting for applications:

Identifies exactly those terms of domain semirings that have the same denotation in all domain semirings

Allows the definition of efficient proof and decision procedures

The domain axioms of domain semirings are the same as for relation algebras and for Kleene algebras with domain

Both relation algebras and Kleene algebras have rich and complex (quasi)equational theories

Rather study the simpler equational theory of domain semirings

P. Jipsen, G. Struth (Chapman U Sheffield) Free Domain Semirings May 23, 2008 12 / 38 Outline of results

Aim: give an explicit description of free domain semirings FDS(X )

First describe free domain monoid FDM(X )

Then show that these elements are the join irreducibles of FDS(X )

⇒ FDS(X ) is isomorphic to the set of finite antichains in the poset of join irreducibles

Show FDS(X ) is representable by a concrete algebra of binary relations, with relational domain as operations ⇒ DS = HSP{Relational domain semirings}

Finally show any distributive lattice with ni -ary operators occurs as domain elements of some domain semiring with ni − 1-ary operators

P. Jipsen, G. Struth (Chapman U Sheffield) Free Domain Semirings May 23, 2008 13 / 38 One-generated domain terms

(D1) d(x)x = x (D2) d(xd(y)) = d(xy) (D3) d(d(x)y)= d(x)d(y)

As usual, we define x0 = 1 and xn+1 = xnx Lemma In a domain monoid, if m ≤ n then

d(xm)xn = xn and d(xm)d(xn)= d(xn)

Proof. Assuming m ≤ n, we write xn = xmxn−m, and using (D1) we have

d(xm)xn = d(xm)xmxn−m = xmxn−m = xn

Now (D3) implies d(xm)d(xn)= d(d(xm)xn)= d(xn)

P. Jipsen, G. Struth (Chapman U Sheffield) Free Domain Semirings May 23, 2008 14 / 38 Expanded normal forms

On elements of the form d(xj ), the order is induced by the meet-semilattice structure: d(xj ) ≤ d(xk ) iff j ≥ k, hence these elements form a chain

Can rewrite any term in expanded normal form:

d(xj0 )xd(xj1 )xd(xj2 )x ··· xd(xjm )

where jk ≥ 1+ jk+1 for k = 0, 1,..., m − 1

E.g. d(x4)x2d(x)= d(x4)xxd(x)= d(x4)xd(x)xd(x)= d(x4)xd(x2)xd(x)

where the last step holds since d(xd(x)) = d(x2) by (D2)

P. Jipsen, G. Struth (Chapman U Sheffield) Free Domain Semirings May 23, 2008 15 / 38 Expanded normal forms

On elements of the form d(xj ), the order is induced by the meet-semilattice structure: d(xj ) ≤ d(xk ) iff j ≥ k, hence these elements form a chain

Can rewrite any term in expanded normal form:

d(xj0 )xd(xj1 )xd(xj2 )x ··· xd(xjm )

where jk ≥ 1+ jk+1 for k = 0, 1,..., m − 1

E.g. d(x4)x2d(x)= d(x4)xxd(x)= d(x4)xd(x)xd(x)= d(x4)xd(x2)xd(x)

where the last step holds since d(xd(x)) = d(x2) by (D2)

P. Jipsen, G. Struth (Chapman U Sheffield) Free Domain Semirings May 23, 2008 15 / 38 Decreasing sequences of numbers

For brevity denote such a term by the sequence (j0, j1, j2,..., jm)

Note that this is always a strictly decreasing sequence of nonnegative integers

Let P = (P, ≤) be the set of all such sequences, ordered by reverse pointwise order

Thus sequences of different length are not comparable, and the maximal elements of this poset are

(0), (1, 0), (2, 1, 0), ...

corresponding to the terms

d(1) = 1, d(x)xd(1) = x, d(x2)xd(x)xd(1) = x2, ...

P. Jipsen, G. Struth (Chapman U Sheffield) Free Domain Semirings May 23, 2008 16 / 38 The poset of join-irreducibles below 1 and x

1= d(x0) (0) x (1, 0) d(x2)x (2, 0) d(x1) (1) d(x3)x (3, 0) (2, 1) = xd(x) d(x4)x (4, 0) (3, 1) = d(x3)xd(x) d(x2) (2) d(x5)x (5, 0) (3, 2) = xd(x2) d(x6)x. (6, 0) (4, 2) = d(x4)xd(x2) .. d(x3) (3) d(x6)xd(x). (6, 1) (4, 3) = xd(x3) .. d(x6)xd(x2). (6, 2) (5, 3) = d(x5)xd(x3) .. d(x4) (4) d(x6)xd(x3). (6, 3) (5, 4) = xd(x4) .. d(x6)xd(x4). (6, 4) .. d(x5) . (5) xd(x5). (6, 5) . ..

P. Jipsen, G. Struth (Chapman U Sheffield) Free Domain Semirings May 23, 2008 17 / 38 The poset of join-irreducibles below x 2

x2 = (2, 1, 0) (3, 1, 0) (4, 1, 0) (3, 2, 1) (5, 1, 0) (4, 2, 1) (6, 1, 0). .. (6, 2, 0). (4, 3, 2) .. (6, 3, 0). (5, 3, 2) .. (6, 4, 0). .. (5, 4, 3) (6, 5, 0). .. , , .. (6 4 3) ......

P. Jipsen, G. Struth (Chapman U Sheffield) Free Domain Semirings May 23, 2008 18 / 38 The product of two decreasing sequences

A multiplication is defined on P by the following “ripple product”

′ ′ ′ ′ (j0, j1, j2,..., jm) · (k0, k1, k2,..., kn) = (j0, j1, j2,..., jm, k1, k2,..., kn)

′ ′ ′ where jm = max(jm, k0) and ji = max(ji , ji+1 + 1) for i = m − 1,..., 2, 1, 0 For example, (7, 3, 2) · (4, 3, 1) = (7, 5, 4, 3, 1),

while (4, 3, 1) · (7, 3, 2) = (9, 8, 7, 3, 2)

Can show that this is the result of multiplying the corresponding expanded normal forms and rewriting result in expanded normal form

It is tedious but not difficult to check that this operation is associative

P. Jipsen, G. Struth (Chapman U Sheffield) Free Domain Semirings May 23, 2008 19 / 38 Domain and partial order

The domain of a sequence (j0, j1, j2,..., jm) is the length-one sequence (j0)

This corresponds to the domain term d(xj0 )

Let A(P) be the set of finite antichains of P

A partial order is defined on A(P) by a ≤ b iff ↓a ⊆ ↓b

The multiplication is extended to antichains by using the complex product (i.e. U · V = {uv : u ∈ U, v ∈ V }) and by removing all non-maximal elements

P. Jipsen, G. Struth (Chapman U Sheffield) Free Domain Semirings May 23, 2008 20 / 38 Representation Theorem

The first result shows that the one-generated free domain semiring can be represented in terms of antichains of decreasing integer sequences Theorem

The join irreducibles of FDS(x) form a poset that is isomorphic to P and FDS(x) is isomorphic to A(P)

Proof. (outline) By distributivity, each domain semiring term t(x) can be written as a finite join of expanded normal form terms

Hence any join irreducible element of FDS(x) can be represented by an expanded normal form term To show that P is the poset of these join irreducible, it suffices to show that all expanded normal forms are join irreducible, and that two expanded normal form terms can be distinguished in some domain monoid

P. Jipsen, G. Struth (Chapman U Sheffield) Free Domain Semirings May 23, 2008 21 / 38 Example terms and corresponding relation

s f

a = (4, 2, 1)

4 2 ta(x)= d(x )xd(x )xd(x)

b = (4, 3, 1) Xa = {arrows} 4 3 tb(x)= d(x )xd(x )xd(x)

Then ta(Xa)= {(s, f )} but tb(Xb)= ∅

P. Jipsen, G. Struth (Chapman U Sheffield) Free Domain Semirings May 23, 2008 22 / 38 Represention of semirings by binary relations

First note that for free idempotent semirings this is always possible [Bredihin, Schein 1978]

For a set X of generators, a concrete construction can be obtained by considering the complex algebra of the free group FGrp(X )

This is always a representable relation algebra, with the elements of the group as disjoint relations

Since the free monoid X ∗ is a subset of the free group, the finite unions of the relations corresponding to singleton words give a relational representation of the free idempotent semiring with X as set of generators

P. Jipsen, G. Struth (Chapman U Sheffield) Free Domain Semirings May 23, 2008 23 / 38 Represention of semirings by binary relations

However, not all idempotent semirings can be represented by ∪, ◦ semirings of relations

[Andreka 1988, 1991] showed that the class of algebras of relations, closed under ∪, ◦, though definable by quasiequations, is not finitely axiomatizable

Hence it is strictly smaller than the finitely based variety of idempotent semirings

Similarly the class of algebras of relations closed under ∪, ∅, ◦, id, d, where d(R)= R;R` ∩ id, is a non-finitely axiomatizable quasivariety, but not a variety

P. Jipsen, G. Struth (Chapman U Sheffield) Free Domain Semirings May 23, 2008 24 / 38 Represention of semirings by binary relations

Theorem The one-generated free domain semiring can be represented by a domain semiring of binary relations

Proof.

(outline) To see that FDS(x) can be represented by a collection of binary relations, with operations of union, composition and domain, it suffices to construct a relation X on a set U such that s(X ) 6= t(X ) in the relation domain semiring P(U × U) for any distinct pair of elements of FDS(x) This is done similarly to the proof of the preceding theorem, by taking X to be the union (over disjoint base sets) of all the relations Xj corresponding to the sequences j ∈ P

P. Jipsen, G. Struth (Chapman U Sheffield) Free Domain Semirings May 23, 2008 25 / 38 n-generated case (briefly)

So far our analysis has considered the one-generated free domain semiring

The n-generated case is more complex, but has recently also been handled

A d-term is any term of the form d(t)

The identities d(x)x = x (D1), d(d(x)y)= d(x)d(y) (D3),

d(d(x)) = x, d(1) = 1, x1= x and 1x = x

form a confluent and terminating rewrite system when applied from left to right (modulo associativity).

Will assume that all terms have been pre-normalized with respect to these identities, so non-constant terms contain no occurrance of 1 or of ... d(d(...) ...) ....

P. Jipsen, G. Struth (Chapman U Sheffield) Free Domain Semirings May 23, 2008 26 / 38 Hence every d-term is of the form

d(vd(t1) ··· d(tn−1)tn) or d(vd(t1) ··· d(tn−1)d(tn))

where v is a variable and the ti are arbitrary pre-normalized terms.

In the first case, the term tn is a variable or a product of terms that starts with a variable, so one application of (D2) will convert it into the second form.

E.g. in the case n = 1, with t1 = xd(y) we have d(vxd(y)) converted to d(vd(xd(y))).

This pre-normalized form for d-terms can be visualized by an edge-labelled tree, where the root has a single outgoing edge labelled v, and attached to the endpoint of this edge there are n (unordered) subtrees representing the d-terms d(t1),..., d(tn)

P. Jipsen, G. Struth (Chapman U Sheffield) Free Domain Semirings May 23, 2008 27 / 38 Let v, w be variables and s1,..., sm, t1,..., tn be terms.

Define a preorder relation ⊑ on pre-normalized d-terms as follows.

d(vd(s1) ··· d(sm)) ⊑ d(wd(t1) ··· d(tn)) iff v = w and ∀j ≤ n∃i ≤ m(d(si ) ⊑ d(tj ))

This relation is obviously reflexive, and it is straight forward to check that it is transitive.

P. Jipsen, G. Struth (Chapman U Sheffield) Free Domain Semirings May 23, 2008 28 / 38 We say that two terms s, t are equal up to d-commutativity if t can be derived from s by a sequence of applications of d(x)d(y)= d(y)d(x) (D4) (which is clearly reversible, hence this notion is an equivalence relation). Lemma If d(s) ⊑ d(t) and d(t) ⊑ d(s) then s and t are equal up to associativity and d-commutativity.

Lemma If d(s) ⊑ d(t) then the identity d(s)d(t)= d(s) holds in all domain monoids.

A d-term d(vd(t1) ··· d(tn) is said to be reduced if d(ti ) 6⊑ d(tj ) for all i 6= j. For example d(xd(x)d(y)) is reduced while d(xd(xd(x)d(y))d(xd(y))) is not.

P. Jipsen, G. Struth (Chapman U Sheffield) Free Domain Semirings May 23, 2008 29 / 38 Theorem Every d-term is equivalent to a reduced d-term, which is unique up to associativity and d-commutativity.

We now define the normal form for arbitrary terms of DM.

A d-sequence is a finite product of d-terms. Note that 1 is considered a d-sequence given by the empty product. Hence any term t can be written in the form t = s0v1s1v2s2 ··· sn−1vnsn

where v1,..., vn are variables in X and s0,..., sn are d-sequences.

The term t is said to be in expanded normal form if for all i < n some d-term d(s) in the d-sequence si satisfies d(s) ⊑ d(vi+1si+1 ··· vnsn). Lemma

Suppose t = s0v1s1v2s2 ··· sn−1vnsn is in expanded normal form. Then d(t)= s0.

P. Jipsen, G. Struth (Chapman U Sheffield) Free Domain Semirings May 23, 2008 30 / 38 We also extend the preorder ⊑ to d-sequences and arbitrary terms as ′ ′ follows. Let d1,..., dm, d1,..., dn be d-terms. Then the ′ ′ ′ d1 ··· dm ⊑ d1 ··· dn iff for all j < n there exists i < m such that di ⊑ dj . In general, for alternations of d-sequences and variables we define

s0v1s1v2s2 ··· sm−1vmsm ⊑ t0w1t1w2t2 ··· tn−1wntn iff m = n, vi = wi and si ⊑ ti for all i = 1,..., n.

Theorem Every term is equivalent to a term in expanded normal form, which is unique up to associativity and d-commutativity.

P. Jipsen, G. Struth (Chapman U Sheffield) Free Domain Semirings May 23, 2008 31 / 38 1

d(x) d(y)

··· ··· ··· ···

Figure: Below 1 in the meet semilattice FDM(x, y) P. Jipsen, G. Struth (Chapman U Sheffield) Free Domain Semirings May 23, 2008 32 / 38 In the one-generated case, reduced d-terms are of the form d(xj ), d-sequences can always be replaced by a single d-term, and the expanded normal form agrees with the one given before:

d(xj0 )xd(xj1 )xd(xj2 )x ··· xd(xjm ).

where jk > 1+ jk+1 for all k = 0,..., m − 1.

In the two-generated case, we note that t = d(x2)d(xy) and s = d(xd(x)d(y)) are both in expanded normal form, and that s ≤ t.

To see that they represent distinct elements in the free algebra, we need to give an example of a domain monoid that distinguishes them.

Observe that the term t(x, y)= d(x2)d(xy) represents a tree with a root 0 and two branches with vertices 0, 1, 2 and 0, 3, 4.

The edges of the first branch are labeled by x and x, while the edges of the second branch are labeled by x and y.

P. Jipsen, G. Struth (Chapman U Sheffield) Free Domain Semirings May 23, 2008 33 / 38 Thus the tree defines a binary relation Xt = {(0, 1), (1, 2), (0, 3)} and another one Yt = {(3, 4)}.

It is easy to check that t(Xt , Yt )= {(0, 0)}, whereas s(Xt , Yt )= ∅.

⇒ Relational representation similar to the one-generated case

Future research is also aiming to describe the structure of free domain semirings in the presence of additional axioms

[Desharnais, Struth 2008] show that the domain algebras d(S) induced by the domain axioms can be turned into (co-)Heyting algebras or Boolean algebras by imposing further constraints

P. Jipsen, G. Struth (Chapman U Sheffield) Free Domain Semirings May 23, 2008 34 / 38 Anti-domain (domain complement)

In particular, adding the three axioms

a(x)x = 0, a(xy) ≤ a(xa(a(y))) and a(a(x)) + a(x) = 1

for an antidomain function a : S → S to the semiring axioms and defining domain as d(x)= a(a(x)) suffices to ensure d(S) is a Boolean algebra

Allows direct encoding of straight-line programs in antidomain semirings:

p;q = p · q and if t then p else q = t · p + a(t) · q

⇒ recover all theorems of the original two-sorted axiomatization of [Desharnais, M¨oller, Struth 2006]

Based on these results, in particular the structure of the free Boolean domain semirings certainly deserve further investigation

P. Jipsen, G. Struth (Chapman U Sheffield) Free Domain Semirings May 23, 2008 35 / 38 Boolean domain semirings generalize J´onsson-Tarski BAOs

|xip = d(xp) is a modal operator on d(A)

In general f is an operator if f (..., x + y,...)= f (..., x,...)+ f (..., y,...) and f (..., 0,...) = 0

BAO = BA with operators B = (B, +, 0, ·, 1, −, (fi )i∈I )

BDSO = Boolean DS with operators A = (A, +, 0, ·, 1, a, (gi )i∈I )

Define d(A) = (a(a(A)), +, 0, ·, 1, a, (|gi i)i∈I ) where

|gi i(p0,..., pn)= a(a(gi (p0,..., pn−1) · pn)) Theorem For any BAO B there exists a Boolean DSO A such that B = d(A)

P. Jipsen, G. Struth (Chapman U Sheffield) Free Domain Semirings May 23, 2008 36 / 38 Domain semirings generalize Gehrke-J´onsson DLOs

DLO = bnd distributive lattices with operators B = (B, +, 0, ·, 1, −, (fi )i∈I )

DSO = domain semirings with operators A = (A, +, 0, ·, 1, d, (gi )i∈I )

Define d(A) = (d(A), +, 0, ·, 1, d, (|gi i)i∈I ) where

|gi i(p0,..., pn)= d(gi (p0,..., pn−1) · pn) Theorem For any DLO B there exists a DSO A such that B = d(A)

A is constructed from the relational domain semiring on the join-irreducibles of the canonical extension of B

Conclusion: Domain semirings give a simple unisorted extension of the static propositional framework to the dynamic framework of sequences

P. Jipsen, G. Struth (Chapman U Sheffield) Free Domain Semirings May 23, 2008 37 / 38 References [H. Andr´eka 1989] On the representation problem of distributive semilattice-ordered semigroups, preprint (1988), Abstracts of the AMS, Vol 10, No 2 (March 1989), p. 174. [H. Andr´eka 1991] Representations of distributive lattice-ordered semigroups with binary relations, Algebra Universalis 28 (1991), 12–25. [G. Birkhoff 1967] “Lattice Theory”, 3rd ed., Vol 25 of AMS Colloquium Publications, AMS, 1967, pp. viii+420. [D. A. Bredihin, B. M. Schein 1978] Representations of ordered semigroups and lattices by binary relations, Colloq. Math. 39 (1978), 1–12. [J. Desharnais, B. M¨oller, G. Struth 2006] Kleene algebra with domain, ACM Transactions on Computational Logic, Vol 7, No 4, 2006, 798–833. [J. Desharnais, G. Struth 2008] Modal Semirings Revisited, Research Report CS-08-01, Department of Computer Science, The University of Sheffield, 2008. [W. McCune 2007] Prover9, www.prover9.org [B. M¨oller, G. Struth 2006] Algebras of modal operators and partial correctness, Theoretical Computer Science, 351, (2006), 221–239.

P. Jipsen, G. Struth (Chapman U Sheffield) Free Domain Semirings May 23, 2008 38 / 38