In this section, we introduce state map filters of semihoops.

#### Definition 5.1

*Let* *H*_{1} *and* *H*_{2} *be semihoops*, *σ* : *H*_{1} → *H*_{2} *be a S*-*map from* *H*_{1} *to* *H*_{2}, *F* *be a filter of* *H*_{1}. *If* *σ*^{−1}(*σ*(*F*)) ⊆ *F*, *we call* *F* *to be a SM*-*filter of* (*H*_{1}, *H*_{2}, *σ*).

#### Example 5.2

*Consider the Example 3.4, one can easily check that the SM*-*filter of* (*H*_{1}, *H*_{2}, *σ*) *are* {*a*_{1}, *b*_{1}, *c*_{1}, 1}, {1_{1}} *and* *H*_{1}.

#### Example 5.3

*Let* *H*_{1} *and* *H*_{2} *be semihoops and* *σ* *be a S*-*map from* *H*_{1} *to* *H*_{2}. *Then* *Ker*(*σ*) = {*x* ∈ *H*_{1}∣*σ*(*x*) = 1_{2}} *is a SM*-*filter of* (*H*_{1}, *H*_{2}, *σ*).

#### Proof

Let *K* = *Ker*(*σ*) and *x*, *y* ∈ *K*. Then *σ*(*x*) = 1_{2} and *σ*(*y*) = 1_{2}. By Proposition 3.6(2) we have *σ*(*x* ⊙_{1} *y*) ≥_{2} *σ*(*x*) ⊙_{2}*σ*(*y*) = 1_{2} ⊙_{21}_{2} = 1_{2}. This means *x* ⊙_{1} *y* ∈ *K*. Let *x* ∈ *K* and *x* ≤ *y*. Then 1_{2} = *σ*(*x*) ≤ *σ*(*y*) and hence *σ*(*y*) = 1_{2}. This shows that *y* ∈ *K*. It follows that *K* is a filter of *H*_{1}. Moreover let *x* ∈ *σ*^{−1}*σ*(*K*). Then *σ*(*x*) ∈ *σ*(K) = {1_{2}} and hence *σ*(*x*) = 1_{2}. Therefore *x* ∈ *K*. This shows that *σ*^{−1}*σ*(K) ⊆ *K*, or *K* is a SM-filter of (*H*_{1}, *σ*). □

#### Definition 5.4

*Let* *H* *be a semihoop and* *σ* *be an IS*-*map on* *H*.

*A filter* *F* *of* *H* *is called state filter of* (*H*,*σ*) *if* *x* ∈ *F* *implies* *σ*(*x*) ∈ *F* *for all* *x* ∈ *H* *[31]*,

*A filter* *F* *of* *H* *is called dual state filter of* (*H*, *σ*) *if* *σ*(*x*) ∈ *F* *implies* *x* ∈ *F* *for all* *x* ∈ *H*,

*A filter* *F* *of* *H* *is called strong state filter of* (*H*, *σ*) *if it is both a state filter and a dual state filter of* (*H*, *σ*).

#### Proposition 5.5

*Let* *H* *be a semihoop and* *σ* *be an IS*-*map on* *H*. *Then each SM*-*filter of* *H* *is a state filter on* *H*.

#### Proof

Let *x* ∈ *F*. Then *σ*(*x*) ∈ *σ* (*F*). Therefore, *σ*(*σ*(*x*)) ∈ *σ*(*F*), that is *σ*(*x*) ∈ *σ*^{−1}(*σ*(*F*)) ⊆ *F*. So *σ*(*x*) ∈ *F*. □

However, the converse of Proposition 5.5 is not true in general.

#### Example 5.6

*Let* *H* = {0, *a*, *b*, 1} *with* 0 ≤ *a*, *b* ≤ 1. *Consider the operation* → *and* ⊙ *as follows*:

$$\overline{)\begin{array}{}\overline{)\begin{array}{ccccc}\odot & 0& a& b& 1\\ 0& 0& 0& 0& 0\\ a& 0& a& 0& a\\ b& 0& 0& b& b\\ 1& 0& a& b& 1\end{array}}\end{array}}\phantom{\rule{1em}{0ex}}\overline{)\begin{array}{}\overline{)\begin{array}{ccccc}\to & 0& a& b& 1\\ 0& 1& 1& 1& 1\\ a& b& 1& b& 1\\ b& a& a& 1& 1\\ 1& 0& a& b& 1\end{array}}\end{array}}$$

*Then* *H* *is semihoop*. *Now, we define* *σ* *follows*: *σ*0 = *a*, *σ* *a* = *a*, *σ* *b* = 1, *σ* 1 = 1. *One can easily check that* *σ* *is an IS*-*map on* *H*. *It is clear that* {*a*, 1} *is a state filter of* (*H*, *σ*), *but it is not a SM*-*filter of* (*H*, *σ*).

#### Proposition 5.7

*Let* *H* *be a semihoop*, *σ* *be an IS*-*map on* *H* *and* *F* ⊆ *H*. *Then the following are equivalent*:

*F* *is a SM*-*filter of* *H*,

*F* *is a strong state filter on* *H*.

#### Proof

(1) ⇒ (2) Let *F* be a SM-filter of (*H*, *σ*). By Proposition 5.5 we only need to prove that *σ*(*x*) ∈ *F* implies *x* ∈ *F*. Let *σ*(*x*) ∈ *F*. Then *σ*(*x*) = *σ*(*σ*(*x*)) ∈ *σ*(*F*). Hence there is *t* ∈ *F* such that *σ*(*x*) = *σ*(*t*). It follows from (1) that *x* ∈ *σ*^{−1}(*σ*(*t*)) ⊆ *σ*^{−1}(*σ*(*F*)) ⊆ *F*. That is *x* ∈ *F*.

(2)⇒ (1) Assume that *F* is a strong state filter on *H*. For *x* ∈ *σ*^{−1}(*σ*(*F*)), we have *σ*(*x*) ∈ *σ*(*F*). Since *F* is strong filter of *H*, we get *x* ∈ *F* and hence *σ*^{−1}(*σ*(*F*)) ⊆ *F*. □

Let *H*_{1} and *H*_{2} be two semihoops and *σ* be a S-map from *H*_{1} to *H*_{2}. For any nonempty set *X* of *H*_{1}, we denote by 〈*X*〉_{σ} the SM-filter of (*H*_{1}, *σ*) generated by *X*, that is, 〈*X*〉_{σ} is the smallest SM-filter of (*H*_{1}, *σ*) containing *X*.

Let *H* be be a semihoop and *σ* be an IS-map on *H*. For any nonempty set *X* of *H*, we denote by 〈*X*〉_{S} (〈*X*〉_{DS}) the state filter (the dual state filter) of (*H*, *σ*) generated by *X*, that is, 〈*X*〉_{S} (〈*X*〉_{DS}) is the smallest state filter (the dual state filter) of (*H*, *σ*) containing *X*.

Denote (*X*)_{DS} = {*x* ∈ *H* ∣ *σ*(*x*) ≥ *x*_{1} ⊙ *σ*(*x*_{1}) ⊙ ⋯ ⊙ *x*_{m} ⊙ *σ*(*x*_{m}), *x*_{i} ∈ *X*}. In the following we discuss the structures of 〈*X*〉_{S}, 〈*X*〉_{DS} and 〈*X*〉_{σ}.

#### Theorem 5.8

*Let* *H* *be a semihoop*, *σ* *be an IS*-*map on* *H* *and* *X* ⊆ *H*. *Then*

〈*X*〉_{S} = {*x* ∈ *H* ∣ *x* ≥ *x*_{1} ⊙ *σ*(*x*_{1}) ⊙ ⋯ ⊙ *x*_{n} ⊙ *σ*(*x*_{n}), *x*_{i} ∈ *X*, *m* ∈ *N*},

(*X*)_{DS} *is a dual state filter of* (*H*, *σ*) *containing* *X*, *and hence* 〈*X*〉_{DS} ⊆ (*X*)_{DS},

〈*X*〉_{σ} = 〈*X*〉_{S} ∪ (*X*)_{DS}.

#### Proof

The proof is similar to that of He et al [30].(Theorem 4.13).

Let *x*, *y* ∈ (*X*)_{DS}. Then *σ*(*x*) ≥ *x*_{1} ⊙ *σ*(*x*_{1}) ⊙ ⋯ ⊙ *x*_{n} ⊙ *σ*(*x*_{n}) for some *x*_{i} ∈ *X*, *n* ∈ *N* and *σ*(*y*) ≥ *y*_{1} ⊙ *σ*(*y*_{1}) ⊙ ⋯ ⊙ *y*_{m} ⊙ *σ*(*y*_{m}) for some *y*_{j} ∈ *X*, *m* ∈ *N*. Hence *σ*(*x* ⊙ *y*) ≥ *σ*(*x*) ⊙ *σ*(*y*) ≥ *x*_{1} ⊙ *σ*(*x*_{1}) ⊙ ⋯ ⊙ *x*_{n} ⊙ *σ*(*x*_{n}) ⊙ *y*_{1} ⊙ *σ*(*y*_{1}) ⊙ ⋯ ⊙ *y*_{m} ⊙ *σ*(*y*_{m}). So *x* ⊙ *y* ∈ (*X*)_{DS}. Assume *x* ≤ *y* and *x* ∈ (*X*)_{DS}. Then *σ*(*y*) ≥ *σ*(*x*) ≥ *x*_{1} ⊙ *σ*(*x*_{1}) ⊙ ⋯ ⊙ *x*_{n} ⊙ *σ*(*x*_{n}) for some *x*_{i} ∈ *X*. It follows that *y* ∈ (*X*)_{DS}. This shows that (*X*)_{DS} is a filter of *H*. Moreover, let *σ*(*x*) ∈ (*X*)_{DS}. Then *σ*(*σ*(*x*)) ≥ *x*_{1} ⊙ *σ*(*x*_{1}) ⊙ ⋯ ⊙ *x*_{n} ⊙ *σ*(*x*_{n}) for some *x*_{i} ∈ *X* and hence *σ*(*x*) ≥ *x*_{1} ⊙ *σ*(*x*_{1}) ⊙ ⋯ ⊙ *x*_{n} ⊙ *σ*(*x*_{n}) for some *x*_{i} ∈ *X*. This shows that *x* ∈ (*X*)_{DS} and hence (*X*)_{DS} is a dual state filter of (*H*, *σ*). Clearly *X* ⊆ (*X*)_{DS}.

Denote *B* = 〈*X*〉_{S} ∪ (*X*)_{DS}. Let *x*, *y* ∈ *B*. If *x*, *y* ∈ 〈*X*〉_{S}, then *x* ⊙ *y* ∈ 〈*X*〉_{S} ⊆ *B* by (1). If *x*, *y* ∈ (*X*)_{DS}, then *x* ⊙ *y* ∈ (*X*)_{DS} ⊆ *B* by (2). Let *x* ∈ 〈*X*〉_{S} and *y* ∈ (*X*)_{DS}. Then *σ*(*x*) ∈ 〈*X*〉_{S} since 〈*X*〉_{S} is a state filter of (*H*, *σ*) by (1). Hence *σ*(*x*) ≥ *x*_{1} ⊙ *σ*(*x*_{1}) ⊙ ⋯ ⊙ *x*_{n} ⊙ *σ*(*x*_{n}) for some *x*_{i} ∈ *X* and *σ*(*y*) ≥ *y*_{1} ⊙ *σ*(*y*_{1}) ⊙ ⋯ ⊙ *y*_{m} ⊙ *σ*(*y*_{m}) for some *y*_{j} ∈ *X* and hence *σ*(*x* ⊙ *y*) ≥ *σ*(*x*) ⊙ *σ*(*y*) ≥ *x*_{1} ⊙ *σ*(*x*_{1}) ⊙ ⋯ ⊙ *x*_{n} ⊙ *σ*(*x*_{n}) ⊙ *y*_{1} ⊙ *σ*(*y*_{1}) ⊙ ⋯ ⊙ *y*_{m} ⊙ *σ*(*y*_{m}) for some *x*_{i},*y*_{j} ∈ *X*. It follows that *x* ⊙ *y* ∈ (*X*)_{DS} ⊆ *B*. Combining the above arguments we get that *B* is closed on ⊙. It is easy to check that if *x* ∈ *B* and *x* ≤ *y* then *y* ∈ *B*. Clearly *X* ⊆ *B*. Now we prove that *B* is a state filter. Let *x* ∈ *B*. If *x* ∈ 〈*X*〉_{S}, then *σ*(*x*) ∈ 〈*X*〉_{S} since 〈*X*〉_{S} is a state filter. If *x* ∈ (*X*)_{DS}, then *σ*(*x*) ∈ 〈*X*〉_{S} ⊆ *B*. So *B* is a state filter. Moreover we prove that *B* is a dual state filter. Let *σ*(*x*) ∈ *B*. If *σ*(*x*) ∈ 〈*X*〉_{S}, then *x* ∈ (*X*)_{DS} ⊆ *B*. Let *σ*(*x*) ∈ (*X*)_{DS}. Then *x* ∈ (*X*)_{DS} since (*X*)_{DS} is a dual state filter by (2). This shows that *B* a dual state filter. By Proposition 5.7, *B* a SM-filter. Let *F* be a SM-filter of (*H*, *σ*) containing *X* and *x* ∈ *B*. If *x* ∈ 〈*X*〉_{S}, then *x* ≥ *x*_{1} ⊙ *σ*(*x*_{1}) ⊙ ⋯ ⊙ *x*_{n} ⊙ *σ*(*x*_{n}) for *x*_{i} ∈ *X*. Since *X* ⊆ *F* and *F* is a SM-filter of (*H*, *σ*), we have *x*_{1} ⊙ *σ*(*x*_{1}) ⊙ ⋯ ⊙ *x*_{n} ⊙ *σ*(*x*_{n}) ∈ *F*. So *x* ∈ *F*. If *x* ∈ (*X*)_{DS}, then *σ*(*x*) ∈ 〈*X*〉_{S} by (1). If *σ*(*x*) ≥ *x*_{1} ⊙ *σ*(*x*_{1}) ⊙ ⋯ ⊙ *x*_{n} ⊙ *σ*(*x*_{n}) for *x*_{i} ∈ *X*. Since *F* is a SM-filter of (*H*, *σ*) containing *X*, then *x*_{1} ⊙ *σ*(*x*_{1}) ⊙ ⋯ ⊙ *x*_{n} ⊙ *σ*(*x*_{n}) ∈ *F* and hence *σ*(*x*) ∈ *F*. Note that *F* is also a dual state filter, we have *x* ∈ *F*. Combining the above arguments we get *B* ⊆ *F*. It follows that *B* = 〈*X*〉_{σ}. □

#### Proposition 5.9

*Let* *H* *be a semihoop*, *σ* *be an IS*-*map and* *F* *be state filters of* (*H*, *σ*) *and* *a* ∉ *F*. *Then*

〈*a*〉_{σ} = {*x* ∈ *H* ∣ *x* ≥ (*a* ⊙ *σ*(*a*))^{n}, *n* ≥ 1} ∪ {*x* ∈ *H* ∣ *σ*(*x*) ≥ (*a* ⊙ *σ*(*a*))^{n}, *n* ≥ 1},

〈*F*, {*a*}〉_{σ} = {*x* ∈ *H* ∣ *x* ≥ *f* ⊙ (*a* ⊙ *σ*(*a*))^{n}, *f* ∈ *F*, *n* ≥ 1} ∪ {*x* ∈ *H* ∣ *σ*(*x*) ≥ *f* ⊙ (*a* ⊙ *σ*(*a*))^{n}, *f* ∈ *F*, *n* ≥ 1},

*if* *a* ≤ *b*, *then* 〈*b*〉_{σ} ⊆ 〈*a*〉_{σ},

〈*a* ⊙ *a*〉_{σ} = 〈*a*〉_{σ},

〈*σ*(*a*)〉_{σ} = 〈*a*〉_{σ},

〈*a* ⊙ *σ*(*a*)〉_{σ} = 〈*a*〉_{σ},

*if* *H* *is a* ⊔-*semihoop, then* 〈*a*〉_{σ} ∩ 〈*b*〉_{σ} = 〈(*a* ⊙ *σ*(*a*)) ⊔ (*b* ⊙ *σ*(*b*))〉_{σ}.

#### Proof

The proofs of (1)–(4) are obvious.

(5) Let *x* ∈ 〈*σ*(*a*)〉_{σ}. Then *x* ≥ (*σ*(*a*) ⊙ *σ*^{2}(*a*))^{n} = (*σ*(*a*))^{2n} ≥ (*a* ⊙ *σ*(*a*))^{2n} or *σ*(*x*) ≥ (*a* ⊙ *σ*(*a*))^{2n} and hence *x* ∈ 〈*a*〉_{σ}. Conversely, let *x* ∈ 〈*a*〉_{σ}. Then *x* ≥ (*a* ⊙ *σ*(*a*))^{n} or *σ*(*x*) ≥ (*a* ⊙ *σ*(*a*))^{n}. Hence *σ*(*x*) ≥ (*σ*(*a*) ⊙ *σ*^{2}(*a*))^{n}. It follows that *σ*(*x*) ∈ 〈*σ*(*a*)〉_{σ}. Since 〈*σ*(*a*)〉_{σ} is a dual state filter we have *x* ∈ 〈*σ*(*a*)〉_{σ}.

(6) Since *a* ⊙ *σ*(*a*) ≤ *a* we have 〈*a*〉_{σ} ⊆ 〈*a* ⊙ *σ*(*a*)〉_{σ} by (3). Conversely, by use of (3), (4) and (5) we have 〈*a* ⊙ *σ*(*a*)〉_{σ} = 〈*σ*(*a* ⊙ *σ*(*a*))〉_{σ} ⊆ 〈*σ*(*a*) ⊙ *σ*^{2}(*a*)〉_{σ} = 〈*σ*(*a*) ⊙ *σ*(*a*)〉_{σ} = 〈 *σ*(*a*)〉_{σ} = 〈*a*〉_{σ}.

(7) Suppose that *H* is a ⊔-semihoop. From *a* ⊙ *σ*(*a*) ≤ (*a* ⊙ *σ*(*a*)) ⊔ (*b* ⊙ *σ*(*b*)), we have that 〈(*a* ⊙ *σ*(*a*)) ⊔ (*b* ⊙ *σ*(*b*))〉_{σ} ⊆ 〈*a* ⊙ *σ*(*a*)〉_{σ} = 〈*a*〉_{σ}. Similarly, we can prove 〈(*a* ⊙ *σ*(*a*)) ⊔ (*b* ⊙ *σ*(*b*))〉_{σ} ⊆ 〈*b*〉_{σ}. Thus, 〈(*a* ⊙ *σ*(*a*)) ⊔ (*b* ⊙ *σ*(*b*))〉_{σ} ⊆ 〈*a*〉_{σ} ∩ 〈*b*〉_{σ}. Conversely, let *x* ∈ 〈*a*〉_{σ} ∩ 〈*b*〉_{σ}. Then there exist *n*, *m*, *s*, *t* ≥ 1, such that *x* ≥ (*a* ⊙ *σ*(*a*))^{n} or *σ*(*x*) ≥ (*a* ⊙ *σ*(*a*))^{n}, and *x* ≥ (*b* ⊙ *σ*(*b*))^{n} or *σ*(*x*) ≥ (*b* ⊙ *σ*(*b*))^{n}. To complete the proof, we divide four cases as following:

Let *x* ≥ (*a* ⊙ *σ*(*a*))^{n} and *x* ≥ (*b* ⊙ *σ*(*b*))^{n}. Then *x* ≥ (*a* ⊙ *σ*(*a*))^{n} ⊔ (*b* ⊙ *σ*(*b*))^{n} ≥ ((*a* ⊙ *σ*(*a*)) ⊔ (*b* ⊙ *σ*(*b*)))^{ns} ≥ (((*a* ⊙ *σ*(*a*)) ⊔ (*b* ⊙ *σ*(*b*))) ⊙ *σ*((*a* ⊙ *σ*(*a*)) ⊔ (*b* ⊙ *σ*(*b*))))^{ns} by Proposition 2.6(3). We deduce that *x* ∈ 〈(*a* ⊙ *σ*(*a*)) ⊔ (*b* ⊙ *σ*(*b*))〉_{σ}.

Let *σ*(*x*) ≥ (*a* ⊙ *σ*(*a*))^{n} and *σ*(*x*) ≥ (*b* ⊙ *σ*(*b*))^{n}. Similarly to (a) we can get *σ*(*x*) ∈ 〈(*a* ⊙ *σ*(*a*)) ⊔ (*b* ⊙ *σ*(*b*))〉_{σ}. Since 〈(*a* ⊙ *σ*(*a*)) ⊔ (*b* ⊙ *σ*(*b*))〉_{σ} is a dual state filter we have *x* ∈ 〈(*a* ⊙ *σ*(*a*)) ⊔ (*b* ⊙ *σ*(*b*))〉_{σ}.

Let *x* ≥ (*a* ⊙ *σ*(*a*))^{n} and *σ*(*x*) ≥ (*b* ⊙ *σ*(*b*))^{n}. Then *σ*(*x*) ≥ (*σ*(*a*) ⊙ *σ*(*a*))^{n} ≥ (*a* ⊙ *σ*(*a*))^{2n} and *σ*(*x*) ≥ (*b* ⊙ *σ*(*b*))^{n}. Thus *σ*(*x*) ≥ (*a* ⊙ *σ*(*a*))^{2n} ⊔ (*b* ⊙ *σ*(*b*))^{n} ≥ ((*a* ⊙ *σ*(*a*)) ⊔ (*b* ⊙ *σ*(*b*)))^{2nt} ≥ (((*a* ⊙ *σ*(*a*)) ⊔ (*b* ⊙ *σ*(*b*))) ⊙ *σ*((*a* ⊙ *σ*(*a*)) ⊔ (*b* ⊙ *σ*(*b*))))^{2nt}. It follows that *σ*(*x*) ∈ 〈(*a* ⊙ *σ*(*a*)) ⊔ (*b* ⊙ *σ*(*b*))〉_{σ}. Since 〈(*a* ⊙ *σ*(*a*)) ⊔ (*b* ⊙ *σ*(*b*))〉_{σ} is a dual state filter we have *x* ∈ 〈(*a* ⊙ *σ*(*a*)) ⊔ (*b* ⊙ *σ*(*b*))〉_{σ}.

Let *σ*(*x*) ≥ (*a* ⊙ *σ*(*a*))^{n} and *x* ≥ (*b* ⊙ *σ*(*b*))^{n}. Similarly to the case (c) we can get *x* ∈ 〈(*a* ⊙ *σ*(*a*)) ⊔ (*b* ⊙ *σ*(*b*))〉_{σ}.

Combining the above arguments we can prove 〈*a*〉_{σ} ∩ 〈*b*〉_{σ} ⊆ 〈(*a* ⊙ *σ*(*a*)) ⊔ (*b* ⊙ *σ*(*b*))〉_{σ}. Therefore 〈*a*〉_{σ} ∩ 〈*b*〉_{σ} = 〈(*a* ⊙ *σ*(*a*)) ⊔ (*b* ⊙ *σ*(*b*))〉_{σ}. □

#### Definition 5.10

*Let* *H*_{1} *and* *H*_{2} *be two semihoops and* *σ* *be a S*-*map from* *H*_{1} *to* *H*_{2}. *A proper SM*-*filter* *F* *of* (*H*_{1}, *H*_{2}, *σ*) *is called a prime SM*-*filter of* (*H*_{1}, *H*_{2}, *σ*), *if for all SM*-*filters* *F*_{1}, *F*_{2} *of* (*H*_{1}, *σ*) *such that* *F*_{1} ∩ *F*_{2} ⊆ *F*, *then* *F*_{1} ⊆ *F* *or* *F*_{2} ⊆ *F*.

Let *H*_{1} and *H*_{2} be two semihoops and *σ* be a S-map from *H*_{1} to *H*_{2}. We denote by PSMF[*H*] the set of all prime SM-filters of (*H*_{1}, *σ*).

#### Example 5.11

*Consider the Example 3.4, one can check that* *F* = {*a*_{1}, *b*_{1}, *c*_{1}, 1_{1}} *is a prime SM*-*filter of* (*H*_{1}, *σ*).

#### Theorem 5.12

*Let* *H* *be a* ⊔-*semihoop*, *σ* *be an IS*-*map and* *F* *be a proper SM*-*filter of* (*H*, *σ*). *Then the following are equivalent*:

*F* *is a prime SM*-*filter of* (*H*, *σ*),

*if* ((*x* ⊙ *σ*(*x*)) ⊔ (*y* ⊙ *σ*(*y*))) ⊙ *σ*((*x* ⊙ *σ*(*x*)) ⊔ (*y* ⊙ *σ*(*y*))) ∈ *F* *for some* *x*, *y* ∈ *H*, *then* *x* ∈ *F* *or* *y* ∈ *F*.

#### Proof

(1) ⇒ (2) Let ((*x* ⊙ *σ*(*x*)) ⊔ (*y* ⊙ *σ*(*y*))) ⊙ *σ*((*x* ⊙ *σ*(*x*)) ⊔ (*y* ⊙ *σ*(*y*))) ∈ *F* for some *x*, *y* ∈ *H*. Then 〈*x*〉_{σ} ∩ 〈*y*〉_{σ} = 〈(*x* ⊙ *σ*(*x*)) ⊔ (*y* ⊙ *σ*(*y*))〉_{σ} ⊆ *F*. Since *F* is a prime SM-filter of (*H*, *σ*), then 〈*x*〉_{σ} ⊆ *F* or 〈*y*〉_{σ} ⊆ *F*. Therefore, *x* ∈ *F* or *y* ∈ *F*.

(2) ⇒ (1) Suppose that *F*_{1}, *F*_{2} ∈ SMF[*L*] such that *F*_{1} ∩ *F*_{2} ⊆ *F* and *F*_{1} ⊈ *F* and *F*_{2} ⊈ *F*. Then there exist *x* ∈ *F*_{1} and *y* ∈ *F*_{2} such that *x*, *y* ∉ *F*. Since *F*_{1}, *F*_{2} are SM-filter of (*H*, *σ*), then *x* ⊙ *σ*(*x*) ∈ *F*_{1} and *y* ⊙ *σ*(*y*) ∈ *F*_{2}. From *x* ⊙ *σ*(*x*), *y* ⊙ *σ*(*y*) ≤ (*x* ⊙ *σ*(*x*)) ⊔ (*y* ⊙ *σ*(*y*)), we obtain (*x* ⊙ *σ*(*x*)) ⊔ (*y* ⊙ *σ*(*y*)) ∈ *F*_{1} ∩ *F*_{2} ⊆ *F* and hence ((*x* ⊙ *σ*(*x*)) ⊔ (*y* ⊙ *σ*(*y*))) ⊙ *σ*((*x* ⊙ *σ*(*x*)) ⊔ (*y* ⊙ *σ*(*y*))) ∈ *F*_{1} ∩ *F*_{2} ⊆ *F*. By (2), we get that *x* ∈ *F* or *y* ∈ *F*, which is a contradiction. Therefore, *F* is a prime SM-filter of (*H*, *σ*). □

#### Definition 5.13

*Let* *H*_{1} *and* *H*_{2} *be two semihoops and* *σ* *be a S*-*map from* *H*_{1} *to* *H*_{2}. *A proper SM*-*filter of* (*H*_{1}, *H*_{2}, *σ*) *is called a maximal SM*-*filter if it not strictly contained in any proper SM*-*filter of* (*H*_{1}, *H*_{2}, *σ*).

#### Example 5.14

*Let* *H*_{1} *and* *H*_{2} *be two semihoops and* *σ* *be a S*-*map from* *H*_{1} *to* *H*_{2} *in Example 3.4*. *One can easily check that* *F* = {*a*_{1}, *b*_{1}, *c*_{1}, 1_{1}} *is a maximal SM*-*filter of* (*H*_{1}, *H*_{2}, *σ*).

#### Proposition 5.15

*Let* *H* *be a bounded* ⊔-*semihoop*, *σ* *be an IS*-*map and* *F* *be a proper SM*-*filter of* (*H*, *σ*). *Then the following are equivalent*:

*F* *is a maximal SM*-*filter of* (*H*, *σ*),

*for any* *a* ∉ *F*, *there is an integer* *n* ≥ 1 *such that* (*σ*(*a*)^{n})^{∗} ∈ *F*.

#### Proof

(1) ⇒ (2) Suppose that *F* is a maximal SM-filter of (*H*, *σ*), and let *a* ∉ *F*. Then 〈*F*,*a*〉_{σ} = *H*, which implies 0 ∈ 〈*F*, *a*〉_{σ}. Then there is *f* ∈ *F* and an integer *n* ≥ 1 such that 0 = *f* ⊙ (*a* ⊙ *σ*(*a*))^{n}. So we have 0 = *σ*(0) ≥ *σ*(*f*) ⊙ *σ*(*a*)^{2n}. Therefore, *σ*(*f*) ≥ (*σ*(*a*)^{2n})^{∗}. Thus, (*σ*(*a*)^{2n})^{∗} ∈ *F*.

(2) ⇒ (1) Let *a* satisfy the condition. Since (*σ*(*a*)^{n})^{∗} ⊙ (*a* ⊙ *σ*(*a*))^{n} ≤ (*σ*(*a*)^{n})^{∗} ⊙ (*σ*(*a*))^{n} = 0 and (*σ*(*a*)^{n})^{∗} ∈ *F*, we obtain 0 ∈ 〈*F*,*a*〉_{σ}, that is, 〈*F*, *a*〉_{σ} = *H*. Therefore, *F* is a maximal SM-filter of (*H*, *σ*). □

#### Proposition 5.16

*Let* *H*_{1} *and* *H*_{2} *be two bounded semihoops and* *σ* *be a S*-*map from* *H*_{1} *to* *H*_{2}.

*If* *F*_{2} *is filter of* *σ*(*H*_{1}), *then* *σ*^{−1}(*F*_{2}) *is a SM*-*filter of* (*H*_{1}, *H*_{2}, *σ*).

*If* *σ* *is an IS*-*map on* *H* *and* *F* *is a maximal filter of* *σ* *H*, *then* *σ*^{−1}(*F*) *is a maximal SM*-*filter of* *H*.

#### Proof

Suppose that *F*_{2} is a filter of *σ*(*H*_{1}). If *x*, *y* ∈ *σ*^{−1}(*F*_{2}), then *σ*(*x*),*σ*(*y*) ∈ *F*_{2}. It follows that *σ*(*x*) ⊙ *σ*(*y*) ∈ *F*_{2}. Since *σ*(*x* ⊙ *y*) ≥ *σ*(*x*) ⊙ *σ*(*y*) and *σ*(*x* ⊙ *y*) ∈ *σ*(*H*_{1}), we have *σ*(*x* ⊙ *y*) ∈ *F*_{2}, that is, *x* ⊙ *y* ∈ *σ*^{−1}(*F*_{2}). Let *x*, *y* ∈ *H*_{1} such that *x* ∈ *σ*^{−1}(*F*_{2}) and *x* ≤ *y*. Then *σ*(*x*) ≤ *σ*(*y*). Since *σ*(*x*) ∈ *F* and *σ*(*y*) ∈ *σ*(*H*_{1}), we can obtain that *σ*(*y*) ∈ *F*_{2}, that is, *y* ∈ *σ*^{−1}(*F*_{2}). Thus, *σ*^{−1}(*F*_{2}) is a filter of *H*_{1}. Note that *σ*(*σ*^{−1}(*x*) = *x* for any *x* ∈ *H*_{1}. Hence *σ*^{−1}(*σ*(*σ*^{−1}(*F*)) = *σ*^{−1}(*F*). Thus *σ*^{−1}(*F*_{2}) is a SM-filter of *H*_{1}.

Now, suppose that *F* is a maximal filter of *σ*(*H*). Let *a* ∉ *σ*^{−1}(*F*), thus *σ*(*a*) ∉ *F*. By the maximality of *F*, there is an integer *n* ≥ 1 such that (*σ*(*a*)^{n})^{∗} ∈ *F* ⊆ *σ*(*H*). Since *σ* ((*σ*(*a*)^{n})^{∗}) = (*σ*(*a*)^{n})^{∗} ∈ *F*, we have (*σ*(*a*)^{n})^{∗} ∈ *σ*^{−1}(*F*). Therefore, *σ*^{−1}(*F*) is a maximal SM-filter of *H*. □

#### Proposition 5.17

*Let* *H* *be a bounded semihoop and* *σ* *be an IS*-*map on* *H* *preserving* ⊙.

*If* *F* *is a SM*-*filter of* (*H*, *σ*), *then* *σ*(*F*) *is a SM*-*filter of* (*σ*(*H*),*σ*).

*If* *F* *is a maximal SM*-*filter of* (*H*, *σ*), *then* *σ*(*F*) *is a maximal SM*-*filter of* (*σ*(*H*),*σ*).

#### Proof

Let *σ*(*x*),*σ*(*y*) ∈ *σ*(*F*), then *x*, *y* ∈ *σ*^{−1}*σ*(*F*) ⊆ *F*. Since *F* is a filter, thus *x* ⊙ *y* ∈ *F* and hence *σ*(*x*) ⊙ *σ*(*y*) = *σ*(*x* ⊙ *y*) ∈ *σ*(*F*). Let *σ*(*x*), *σ*(*y*) ∈ *σ*(*H*) such that *σ*(*x*) ∈ *σ*(*F*) and *σ*(*x*) ≤ *σ*(*y*). Since *σ*(*x*) ∈ *σ*(*F*) we have *x* ∈ *σ*^{−1}*σ*(*F*) ⊆ *F*. So *x* ∈ *F*. By Proposition 5.7 we have *σ*(*x*) ∈ *F*. Since *σ*(*x*) ≤ *σ*(*y*) we get *σ*(*y*) ∈ *F*. Using Proposition 5.7 again we obtain *y* ∈ *F*, and so *σ*(*y*) ∈ *σ*(*F*). Thus, *σ*(*F*) is a filter of *σ*(*H*). Now let *x* ∈ *σ*(*F*). Then *x* = *σ*(*t*) for some *t* ∈ *F* and hence *σ*(*x*) = *σ*^{2}(*t*) = *σ*(*t*) = *x* ∈ *σ*(*F*). It follows that *σ*(*F*) is a state filter of (*H*, *σ*). Let *x* ∈ *σ*(*H*) and *σ*(*x*) ∈ *σ*(*F*). Then *x* = *σ*(*t*) for some *t* ∈ *H*. Hence *x* = *σ*(*t*) = *σ*^{2}(*t*) = *σ*(*σ*(*t*)) = *σ*(*x*) ∈ *σ*(*F*). This means that *σ*(*F*) is a dual state filter of (*σ*(*H*),*σ*). Therefore *σ*(*H*) is a strong state filter of (*σ*(*H*),*σ*). By Proposition 5.7 we have that *σ*(*F*) is a SM-filter of (*σ*(*H*),*σ*).

Now, let *F* be maximal and *σ*(*a*) ∉ *σ*(*F*). Then *a* ∉ *F*, and there is an integer *n* ≥ 1 such that (*σ*(*a*)^{n})^{∗} ∈ *F* and hence *σ* ((*σ*(*a*)^{n})^{∗}) = (*σ*(*a*)^{n})^{∗} ∈ *σ*(*F*). Since *σ*(*σ*(*a*)^{n}) ≥ (*σ**σ*(*a*))^{n} = (*σ*(*a*))^{n}, we have (*σ*(*a*)^{n})^{∗} ≥ *σ* ((*σ*(*a*)^{n})^{∗}). Hence (*σ*(*a*)^{n})^{∗} ∈ *σ*(*F*). Therefore, *σ*(*F*) is a maximal SM-filter of (*σ*(*H*),*σ*). □

#### Corollary 5.18

*Let* *H* *be a bounded semihoop and* *σ* *be an IS*-*map on* *H*.

*If* *F* *is a (maximal) filter of* *σ*(*H*), *then* *σ*^{−1}(*F*) *is a strong state (maximal)filter of* (*H*, *σ*).

*If* *σ* *is preserving* ⊙ *and* *F* *is a strong state (maximal) filter of* (*H*, *σ*), *then* *σ*(*F*) *is a strong state (maximal) filter of* (*σ*(*H*),*σ*).

Now, we introduce two kinds of semihoops and give some characterizations of them.

#### Definition 5.19

*Let* *H* *be a semihoop and* *σ* : *H* → *H* *be an IS*-*map on* *H*. *If* (*H*, *σ*) *has exactly one maximal SM*-*filter, we call* (*H*, *σ*) *to be state local*.

#### Theorem 5.20

*Let* *H* *be a semihoop and* *σ* *be an IS*-*map on* *H*. *Then the following are equivalent*:

(*H*, *σ*) *is state local**;*

*σ*(*H*) *is local*.

#### Proof

(1) ⇒ (2) Let *F* be the only maximal SM-filter of (*H*, *σ*). We prove that *σ*(*F*) is the only maximal filter of *σ*(*H*). First, *σ*(*F*) is a proper filter of *σ*(*H*). In fact, if *σ*(*F*) = *σ*(*H*), then 0 ∈ *σ*(*F*), which implies 0 ∈ *F*, a contradiction. Now, let *G* be a filter of *σ*(*H*), *G* ≠ *σ*(*H*) and let *x* ∈ *G*. It follows from Corollary 5.18(1) that *σ*^{−1}(*G*) is a SM-filter of (*H*, *σ*). Thus *σ*^{−1}(*G*) is a proper SM-filter of (*H*, *σ*). Moreover, if *σ*^{−1}(*G*) = *H*, then 0 ∈ *σ*^{−1}(*G*), so 0 ∈ *G*, a contradiction. It follows that *σ*^{−1}(*G*) ⊆ *F*. if *x* = *σ*(*x*) ∈ *G*, then *x* ∈ *σ*^{−1}(*G*), it follows that *x* ∈ *F*. But *x* = *σ*(*x*), so *x* ∈ *σ*(*H*). Thus *G* ⊆ *σ*(*F*). Hence *σ*(*G*) is the only maximal filter of *σ*(*H*). Therefore, *σ*(*H*) is local.

(2) ⇒ (1) Suppose that *G* is the only maximal filter of *σ*(*H*). By Corollary 5.18(1), we have that *σ*^{−1}(*G*) is a maximal SM-filter of (*H*, *σ*). We will prove that *σ*^{−1}(*G*) is the only maximal SM-filter of (*H*, *σ*). Let *G* be a SM-filter of (*H*, *σ*), *F* ≠ *L*. Then *σ*(*F*) is a proper filter of *σ*(*H*), so *σ*(*F*) ⊆ *G*. Let *x* ∈ *F* then *σ*(*x*) ∈ *σ*(*F*) ⊆ *G*. Thus, *x* ∈ *σ*^{−1}(*G*). It follows that *F* ⊆ *σ*^{−1}(*G*). Therefore, (*H*, *σ*) is state local. □

#### Definition 5.21

*Let* *H* *a be semihoop and* *σ* : *H* → *H* *be an IS*-*map on* *H*. *If* (*H*, *σ*) *has two SM*-*filters* {1} *and* *H*, *we call* (*H*, *σ*) *to be simple*.

#### Theorem 5.22

*Let* *H* *a be semihoop and* *σ* : *H* → *H* *be an IS*-*map on* *H* *such that* *σ* *preserving* ⊙ . *Then the following are equivalent*:

(*H*, *σ*) *is simple**;*

*σ*(*H*) *is simple and* *Ker*(*σ*) = {1}.

#### Proof

(1) ⇒ (2) Let *F* be a filter of *σ*(*H*) and *F* ≠ {1}. It follows from Corollary 5.18(1) that *σ*^{−1}*F* is a SM-fiter of (*H*, *σ*). Since (*H*, *σ*) is state simple, we have that *σ*^{−1}(*F*) = {1} or *σ*^{−1}(*F*) = *H*. Notice that *F* ⊆ *σ*^{−1}*F* (if *x* ∈ *F*, then *σ* *x* = *x*, that is, *x* ∈ *σ*^{−1}*F*, we obtain that *σ*^{−1}F ≠ {1}. Thus, *σ*^{−1}F = *H*. Then 0 ∈ *σ*^{−1}*F*, that is, 0 = *σ* 0 ∈ *F*. So we obtain that *F* = *σ* *H*. Therefore, *σ* *H* is simple.

By Example 5.3 we have *Ker*(*σ*) is a SM-filter of (*H*, *σ*) and *Ker*(*σ*) ≠ *H*. It follows that *Ker*(*σ*) = {1}.

(2) ⇒ (1) Let *F* be a SM-filter of (*H*, *σ*) and *F* ≠ {1}. By Corollary 5.18(2), we obtain that *σ* *F* is a filter of *σ* *H*. Since *σ* *H* is simple, we obtain that *σ* *F* = {1} or *σ* *F* = *σ* *x*. Since *Ker*(*σ*) = {1}, we have *F* ≠ {1}. Thus, *σ* *F* = *σ* *x*. Then 0 ∈ *σ* *F*, that is, 0 ∈ *F*. It follows that *F* = *H*. Therefore (*H*, *σ*) is state simple. □

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