Semianneau

In Mathematical, a semi-ring , or a half-ring , is a algebraic Structure $(E, +, \ times, 0,1)$ such as

$(E, +, 0)$ constitutes a commutative Monoïde ;
$(E, \ times, 1)$ form a Monoid ;
$\ times$ is distributive compared to +;
0 is absorbent for the product, in other words: for all $x \ in E: X \ times 0 = 0 \ times X = 0$ .

A half-ring is commutative when its product is commutative.

Note: a ring builds in a similar way starting from a additive group and of a multiplicative Monoïde but it does not need the last axiom because the absorption of the zero results from the inversibility of the sum.

Fields of predilection

The half-rings are often found in:

operations research: the graphs have weights in a half-ring; the product is associated with the accumulation of value along a way and the sum corresponds the made-to-order to compose several ways;
theory of the languages and the automats: the concatenation of (whole of) the chains to manufacture others of them is the product and the union of (whole of) the chains is the sum.

Examples

the simplest half-ring is that of the Boolean ones: $(\ {0, 1 \}, \ vee, \ wedge, 0,1)$ where $\ vee$ and $\ wedge$ is OR and AND respectively.
most natural is perhaps that of the positive entireties with the addition and the multiplication: $(\ mathbb {NR}, +, \ times, 0,1)$ .
the whole of the natural entireties extended to $\ infty$ in a usual way (any sum with $\ infty$ gives $\ infty$ ; very produced with $\ infty$ gives $\ infty$ , except for 0 which remains absorbent) provided with the operator min and of the sum is a half-ring: $(\ mathbb {NR} \ cup \ {\ infty \}, min, +, \ infty, 0)$ is known under the name of tropical half-ring ; it is in the middle of the calculation algorithms of shorter way in a graph: the weights are added along the ways and in front of several ways, one takes the minimal cost.
$(\ mathbb {NR} \ cup \ {\ infty \}, max, min, 0, \ infty)$ is the half-ring subjacent with the calculation of the maximum flow of a graph: in a sequence of arcs, that of minimal weight imposes its flow and in front of several sequences, one takes maximum flow.
the whole of the parts of a unit E provided with the union and intersection is a semi-ring. The two laws are distributive one compared to the other, the neutral element of the union is the empty set, that of the intersection is the unit E. the two laws is commutative and forms with E the two necessary monoids. It is a Boolean algebra and thus a lattice.
All distributive lattice is a semi-ring.

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